If $a+b+c=1$ and a,b,c ＞0 prove $\dfrac{b^2}{a+b^2}+\dfrac{c^2}{b+c^2}+\dfrac{a^2}{c+a^2} \geqslant \dfrac{3}{4}$
inequality with three variables and condition
folder content doesn't get refreshed automatically in ubuntu 14.04 I'm using Ubuntu 14.04 and when I'm doing experiments which creates new files inside a particular folder X. But I can't see these new files inside X without refreshing X.  I have increased the count to 1M in  /proc/sys/fs/inotify/max_user_watches   but still the issue is not fixed.   How to fix it?  EDIT As suggested, output of cat /etc/xdg/autostart/user-dirs-update-gtk.desktop  [Desktop Entry] Encoding=UTF-8 Exec=xdg-user-dirs-gtk-update Name=User folders update Comment=Update common folders names to match current locale Terminal=false OnlyShowIn=GNOME;LXDE;Unity; Type=Application StartupNotify=false X-KDE-autostart-after=panel NoDisplay=true X-Ubuntu-Gettext-Domain=xdg-user-dirs-gtk How does Java's PreparedStatement work? I have a lidar file (file.las) with the following spatial reference:  Spatial Reference:            PROJCS["NAD83 / UTM zone 11N",     GEOGCS["NAD83",         DATUM["North_American_Datum_1983",             SPHEROID["GRS 1980",6378137,298.2572221010002,                 AUTHORITY["EPSG","7019"]],             AUTHORITY["EPSG","6269"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4269"]],     PROJECTION["Transverse_Mercator"],     PARAMETER["latitude_of_origin",0],     PARAMETER["central_meridian",-117],     PARAMETER["scale_factor",0.9996],     PARAMETER["false_easting",500000],     PARAMETER["false_northing",0],     UNIT["metre",1,         AUTHORITY["EPSG","9001"]],     AUTHORITY["EPSG","26911"]]   Then I reproject it to WGS84 with las2las:  las2las --a_srs EPSG:26911 --t_srs EPSG:4326 -i file1.las -o output.las   And I get:   Spatial Reference:                GEOGCS["WGS 84",         DATUM["WGS_1984",             SPHEROID["WGS 84",6378137,298.257223563,                 AUTHORITY["EPSG","7030"]],             AUTHORITY["EPSG","6326"]],         PRIMEM["Greenwich",0],         UNIT["degree",0.0174532925199433],         AUTHORITY["EPSG","4326"]]   But when I try to revert the transformation: from WGS84 to NAD83/UTM 11N, I get an error:  las2las --a_srs EPSG:4326 --t_srs EPSG:26911 -i output.las -o wgs2utm.las ERROR 1: latitude or longitude exceeded limits error: Could not project point for ReprojectionTransform::latitude or longitude exceeded limits0.   Why I got this error, if I am just reversing the transformation (originally from NAD83 to WGS84, and then from WGS84 to NAD83)?. How do I fix gaps around my lean-to greenhouse? reduce Screen brightness in battery mode automatically KDE and XFCE4 has a program to adjust the screen brightness automatically in battery mode to a preset value. How can i set the same with Unity ( or Gnome3 )? Sequence of triangles with exponentiated sides Let $a,b,c\in\Bbb R^+$. For any given $n\in\Bbb N$ there exists a triangle with sides $a^n,b^n,c^n$. Show that at least two of $a,b,c$ are equal.   I know that you can create inequalities for $a^n,b^n,c^n$, but what's next? What does "in spades" mean? What does "in spades" mean, for example in the following sentence:     demand and love are both there in spades ...   I guess "in spades" means "on cards" or "on the table" or exist? Encrypt home drive after installation I declined to encrypt to my home drive whilst installing ubuntu.  Is it possible to encrypt it after installation?  if so, how? Suppose someone from "Harry's world" married a Muggle, and they want to escort their child. Can the Muggle parent get through to the Hogwarts platform? US doctor working in Tripoli wants me to open online account Value of Vandermonde type determinant How do you make corrections to a published paper? What is the "?" symbol in URL used for in php? Is it legal to develop a game using D&D rules? For a while now I've been thinking about trying my hand at creating a game similar in spirit and execution to Baldur's Gate, Icewind Dale and offshoots. I'd rather not face the full bulk of work in implementing my own RPG system - I'd like to use D&amp;D rules.  Now, reading about the subject it seems there is something called "The License" which allows a company to brand a game as D&amp;D. This license seems to be exclusive, and let's just say I don't have the money to buy it :p.  Is it still legal for me to implement and release such a game? Commercially or open-source? I'm not sure exactly which edition would fit the best, but since Baldur's Gate is based of 2nd edition, could I go ahead an implement that?  in short: what are the issues concerning licensing and publishing when it comes to D&amp;D?  Also: Didn't see any similar question... Bibliography with page numbers In GKP's Concrete Mathematics, there is a useful and interesting feature I would like to see in more books. The bibliography entries include:   a - the page numbers where the book or article is cited in the document. b - a few lines of review of the cited work.   It is a kind of mix between bibliography, index and review.  Are there stable LaTeX packages/BibTeX styles to do (a) and (b) ?  EDIT: I wrote (b) initially as "comments about the work cited". It could be understood as "gathering comments made in the book about the work and reproducing them in the bibliography".  This is not what I meant but this could be something useful for others. Let $\sim$ denote the equivalence relation on the $n$-sphere $S^n$ defined via   $$ (x_1,\dots,x_n,x_{n+1})\sim(x_1,\dots,x_n,−x_{n+1})\:\text{ for all }\: (x_1,\dots,x_{n+1})\in S^n. $$      Show that the quotient space $S^n/\sim$ is homeomorphic to the $n$-disc $D^n$.   I know that Ii have to show that there exists a bijective function $f: (S^n/\sim) \to D^n$ that is continuous and that the pre-image $f^{-1}$ is continuous as well, but I can't advance from here, any tips? There is the next characterization for valuation domains:  Let $D$ be an integral domain and $K$ its field of fractions. The following are equivalent:   For every nonzero $x$ in $K$, either $x$ in $D$ or $x^{−1}$ in $D$. The ideals of $D$ are totally ordered by inclusion. The principal ideals of $D$ are totally ordered by inclusion (i.e., the elements in D are totally ordered by divisibility.) There is a totally ordered abelian group $Γ$ (called the value group) and a surjective group homomorphism (called the valuation) $v :$ $K^×$ $→ Γ$ with $D = \{ x ∈$ $K^×$ | $v(x)$ ≥ $0 \}$ $∪$ {0}.   The equivalence between 1, 2 and 3 are clear, and I understand that the fourth can be found in Krull's article (1936), but I don't speak german and I can't find a proof anywhere. Any link or hint will be much appreciated. How can I run a specific command for each find result? How would one run a specific command for each file that was found by using the find command? For the purpose of the question lets say that I would simply like to delete each file found by find. Primary ideals in Noetherian rings For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary or $(0)$ is a primary submodule.  Now for an ideal $\mathfrak{q} \subset R$ this definition gives that $\mathfrak{q}$ is $\mathfrak{p}$-primary iff $\text{Ass}(R/\mathfrak{q}) = \{\mathfrak{p}\}$. It can be shown that this is equivalent with $\mathfrak{p}^n \subset \mathfrak{q}$ for some $n\in \mathbb{N}$ and the condition \begin{align*} \forall r,s \in R: rs \in \mathfrak{q}, r\not\in \mathfrak{p}\Rightarrow s \in \mathfrak{q}. \end{align*}     Now I suspect that if $R$ is Noetherian, and if $\mathfrak{q}$ is $\mathfrak{p}$-primary, then we have $\sqrt{\mathfrak{q}} = \mathfrak{p}$. I have shown this for PID's, but does this generaly hold?    And if so, is every ideal $\mathfrak{q}\subset R$ then $\sqrt{\mathfrak{q}}$-primary?   Many thanks. Is Unit Testing worth the effort? I am working to integrate unit testing into the development process on the team I work on and there are some sceptics. What are some good ways to convince the sceptical developers on the team of the value of Unit Testing? In my specific case we would be adding Unit Tests as we add functionality or fixed bugs. Unfortunately our code base does not lend itself to easy testing.