Can't render fire effect on transparent background
Render semi-transparent volume with cycles
I was trying to solve the following:  Given: $0 &lt; a &lt; b$ and $n&gt;1$, prove: $$na^{n-1}(b-a) &lt; b^n-a^n &lt; nb^{n-1}(b-a)$$  I managed to get this far using the mean value theorem: $$a^n(b-a)&lt;b^n -a^n&lt;b^n (b-a)$$ Any idea how to continue? The API Sandbox This is the same concept as .  Why do we need more than one? this is in order to test the API, some methods support vectored ids (for example /questions/id1;id2/comments) and for that two sandboxes are needed. One of the most commonly asked questions of beginners. How do ranks work in Go for amateurs?  Why do we start backwards? And why do we suddenly count upwards again? What is the difference between 9d and 9p? How is this related to rating? What are the basic differences between ranks used in different countries and different servers? Real Analysis - Prove a set is countable I am asked to prove the following question:  For each finite set of real numbers, $F$, define $\sum F$ to be the sum of all the numbers in $F$.  For each set $P$ of positive real numbers, define $\sum P=\sup\{\sum F: F\text{ is a finite subset of } P\}$.  Suppose $P$ is a set of positive real numbers such that $\sum P &lt; \infty$. Prove that $P$ is a countable set.  My attempt (just a sketch proof):  Suppose $P$ is uncountable, $\exists \epsilon_1 &gt; 0$, s.t. $$ P_1 = [\inf P + \epsilon_1, \sup P]\cap P $$ is uncountable. Pick $a_1 \in [\inf P, \inf P + \epsilon_1] \cap P \in P$, such a $P$ must exists due to greatest lower bound property.  Now $P_1$ is uncountable, $\exists \epsilon_2 &gt; 0$, s.t. $$ P_2 = [\inf P_1 + \epsilon_2, \sup P_1]\cap P_1  $$  is uncountable. Pick $a_2 \in [\inf P_1, \inf P_1 + \epsilon_2]\cap P_1 \in P$. Carry on this process (this should be formalized using induction), we can find  $$ a_1 \leq a_2 \leq a_3 \leq ... $$ with $a_n &gt; 0$ and $a_n \in P$ for all $n$.  Now $\sup\{\sum F_n: F_n = \{a_1,...,a_n\}, n \in \mathbb{N}^+\} = \lim_{n \to \infty}\sum_{i = 1}^n a_i &gt; \lim_{n \to \infty}na_1 = \infty$. This means that $\sum P = \infty$ since $\{\sum F_n: F_n = \{a_1,...,a_n\}, n \in \mathbb{N}^+\} \subset \{\sum F: F\text{ is a finite subset of } P\}$.  I think my proof is a bit clumsy but is it correct? Inheritance of static members in PHP In PHP, if a static attribute is defined in the parent class, it cannot be overridden in a child class. But I'm wondering if there's any way around this.   I'm trying to write a wrapper for someone else's (somewhat clunky) function. The function in question can be applied to lots of different data types but requires different flags and options for each. But 99% of the time, a default for each type would suffice.  It would be nice if this could be done with inheritance, without having to write new functions each time. For example:  class Foo {     public static $default = 'DEFAULT';      public static function doSomething ($param = FALSE ) {         $param = ($param === FALSE) ? self::$default : $param;         return $param;     } }  class Bar extends Foo {     public static $default = 'NEW DEFAULT FOR CHILD CLASS'; }  echo Foo::doSomething() . "\n";  // echoes 'DEFAULT'  echo Bar::doSomething() . "\n";  // echoes 'DEFAULT' not 'NEW DEFAULT FOR CHILD CLASS'  // because it references $default in the parent class :( $ p(x)\in\mathbb{R[X]} $ is a polynomial of degree $n$ with no real   roots. Show that: $$\int\limits_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n^{3/2}\pi.$$   It's easy to see that the degree of $ p$ has to be even. For $n=2$ this integral is at most $2\pi$. For $n&gt;2$ the maximum value of this integral is obtained when all the imaginary parts of the roots of $p(x)$ tend to $0$, but I couldn't go further.    Any help would be appreciated, thanks.    Edits by David Speyer: It seems very likely now that the optimum bound is $n \pi$, not $n^{3/2} \pi$. As pointed out in the comments below, and further in , this is the value we get in the limit where $p$ has double roots on the real axis.   It seems likely that  moving the roots of $p$ away from the real axis can only make the integral less. Write the roots of $p$ as $a_j \pm i b_j$, so $p(x) = \prod ((x-a_j)^2 + b_j^2)$ and  $$\frac{p'(x)}{p(x)} = \sum \frac{2 (x-a_j)}{(x-a_j)^2 + b_j^2}.\quad (\ast)$$  So making the $b_j$ larger tends to make $p'(x)/p(x)$ smaller, which makes the integral smaller. But this argument is not rigorous, because the terms of $(\ast)$ can have both positive and negative sign, so it could be that making the individual terms closer to $0$ makes the absolute value of $(\ast)$ larger.  I don't see how to beat this issue easily.  Thus, I'm putting up a bounty for proving or disproving   $$ \int_{-\infty}^{+\infty}\dfrac{(p'(x))^2}{(p'(x))^2+(p(x))^2}\,dx \leq n\pi.$$ Is there a functional difference between the modern police station and classic police station? Share wireless Internet connection through ethernet I want to share the wireless Internet connection on my desktop with my old laptop, through ethernet.  I have ticked the option "Make available to others" on both connections, on my desktop, but my laptop doesn't connect. Am I doing something wrong or is it a bug? "spent a lot of time to shop" vs. "spent a lot of time shopping" She spends a lot of time to shop. She spends a lot of time shopping.   Are both of these sentences grammatically correct and do they have the same meaning? How to dismiss the Windows 8.1 Upgrade banner? As of today, my Windows 8 system is displaying a large banner covering about a quarter of the screen, demanding that I upgrade to Windows 8.1 for free.  The banner helpfully informs me that that I can continue to use the computer while the upgrade is downloading.  There's a button to go to the store for the upgrade, but no other choices, such as No Thanks, Maybe Later, or Absolutely Not.    I can't find a way to dismiss the banner, and I can start programs but can't give them focus while the banner is present.  So I'm dead in the water.  How can I fix this?  Bringing up Task Manager doesn't give me any options.    My employer doesn't yet support Windows 8.1 (which is also an issue, but beyond my control) so I won't be able to connect to work if install the upgrade.  So there is no way on God's green earth that I'm going to install this upgrade now.    UPDATE: The banner went away after the second restart and I was able to run regedit as @magicandre suggested.  The banner hasn't reappeared.  So far, so good. Word for sentences that make sense when read backwards? Is there a word for sentences that make sense when read normally or backwards (not necessarily by reversing the words)? Example:     "Are you as bored as I am?" Roots in a polynomial ring Can a hyphen be used without anything on the right side? Check if a class is derived from a generic class As an example I have a BSB file from NOAA which is a navigational map. The map has a SRS of WGS84 and a Mercator projection.  I initially translate the file from BSB format to GTiff format using the following command:  gdal_translate -of GTiff 18554_1.kap noaa.tif  I can view this GTiff file using Paint Shop Pro quite fine. I then translate the file from GTiff format to VRT format using the following command:  gdal_translate -of vrt –expand rgba noaa.tif noaa.vrt  I then try to generate a tile matrix set using the following command:  gdal2tiles.py noaa.vrt tiles_folder  Where the map tiles are located in tiles_folder. If I try to open the openmaps.html or google.html, the map zoom in and out looks good. However, on tiles generator like MBTiles, I see tiles swap. See the image below. I am pulling my hair off to get the tiles to line up. I am guessing it is projection error. I am not understanding those projection types. It would great someone can explain it while finding the error. I am planning to use on a tiling  like this. By the way, on GDAL how do I specify the BSB (reference) file that is associated with KAP (data) file? I have seen cases where only one BSB file referred by multiple KAP files. Why is the ROC curve two-dimensional instead of three-dimensional? When I understand it correctly, a  curve composes of a lot of ratios of TPR and FPR. Each such ratio is gained, e.g., by scanning/investigating a parameter where the ratio is the result of the used classifier. In other words: I'm interested in how a classifier separates data points according to a frequency interval of 0-100 Hz. Hence I scan this frequency, record all the 101 ratios, put them into the plot and the ROC curve represents these datapoints, finally.  Wouldn't it be much more interesting when the frequency would be used as a third axis? Or is it implicit in the axis? But anyway, I think it would be easier to look at with a third axis? (I am not asking how to block my posts on my timeline.)  I am asking how I can block my activity, such as posting in open groups, making comments on people's posts, adding a friend, liking a photo, liking a post, etc., from people seeing it in their news feeds?  I don't want people seeing my interactions with other people. What is the best way to add options to a select from a JavaScript object with jQuery? I am trying to prove:     $a\mid b \land b\mid c \Rightarrow a\mid c $   $a\mid b$ means that a divides b if there is an integer k, that $b=k\cdot a$  Please give me a hint on how to start, because I have no idea. combinatorics circular arrangement problem If $n$ distinct things are arranged in a circle, then what are the number of ways selecting three of these things so that no two of them are next to each other?