What do Cyan/Black Edges mean?
What are the light blue lines on my imported objects
I am trying to figure out how to make the space around the sun glow and than fade away into darkness.  The sun was made in blender using cycles render and several noise textures. The glow I have around my sun was made using Blenders Gurus tutorial for earth, atmosphere effect, to create a red glow. The sun will be animated so using composition is not going to work for me (sadly);  I realized one of the reasons why the method is not working is I am using surface rather than a volume shader, but As I tried to do it using volume shaders, I could not figure out what direction to take. Why is Euclid's proof on the infinitude of primes considered a proof? I've expressed Euclid's proof on the infinitude of primes on Mathematica:  f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]   Which results in:  $\begin{array}{ll}  3 &amp; \text{True} \\  7 &amp; \text{True} \\  31 &amp; \text{True} \\  211 &amp; \text{True} \\  2311 &amp; \text{True} \\  30031 &amp; \text{False} \\  510511 &amp; \text{False} \\  9699691 &amp; \text{False} \\  223092871 &amp; \text{False} \\  6469693231 &amp; \text{False} \\  200560490131 &amp; \text{True} \\  7420738134811 &amp; \text{False} \\  304250263527211 &amp; \text{False} \\  13082761331670031 &amp; \text{False} \\  614889782588491411 &amp; \text{False} \\  32589158477190044731 &amp; \text{False} \\  1922760350154212639071 &amp; \text{False} \\  117288381359406970983271 &amp; \text{False} \\  7858321551080267055879091 &amp; \text{False} \\  557940830126698960967415391 &amp; \text{False} \\ \end{array}$  The proof flaws for all those values, how is it considered a proof then? I guess that there might be infinite prime numbers according to the proof, but what is the guarantee that at some point it won't fail indefinitely? Proving $\sqrt{ab} = \sqrt a\sqrt b$ I am currently in high school and we are studying radicals. I had asked my math teacher why $\sqrt{ab}=\sqrt{a}\sqrt{b}$ (for all a,b>0) and he tries to prove it by arguing that $a^{1/2}*b^{1/2}=(ab)^{1/2}$ (an exponent law). However, I find this proof problematic since $x^{1/2}$ is simply defined as $\sqrt{x}$, so the reasoning is circular.   My view is that $\sqrt{ab}=\sqrt{a}\sqrt{b}$ because once we square both sides we get $ab=ab$. Since we're obviously referring to the positive root, and the function $f(x) = \sqrt{x}$ is injective, it necessarily follows that the original expressions $\sqrt{ab}$ and $\sqrt{a}\sqrt{b}$ are equivalent because for injective functions it is not possible to map distinct elements in the domain to the same element in the range ($ab$). Hence they are equivalent expressions.   My question therefore is, is my proof valid and/or rigorous (I find it convincing but maybe it's wrong; I just want to be clear) and secondly was my teacher's proof correct? How to place a .png texture with transparency over a material I am trying to create a low poly face with a face from a .png file. The issue is I already have material placed for hair and skin, as shown in the picture below.   I have created a .png file with a face which I would like to overlay over the skin material on the front of the head.    All examples I searched so far assume I also want the color of the material from the .png texture.  Update:  I added UVMapping, changed view to texture, made sure one face is on the eye but nothing is displayed..... see image... I don't know what is going on. Can't install experimental modules Can the JVM recover from an OutOfMemoryError without a restart Can the program counter on Intel CPUs can be read directly (that is without 'tricks') in kernel mode or some other mode? Referring to unidimensional motion, it is obvious that it doesn't always make sense to write the speed as a function of position. Seems to me that this is a necessary condition to derive formulas like:  $$v^2=v_0 ^2 +2\int_{x_0}^{x}a\cdot dx$$  In fact, in the first step of the demonstration (the one I saw, but I think that this step is crucial) it's required to write $a=dv/dt=(dv/dx)(dx/dt)$, that doesn't make sense if $v$ isn't a function of $x$.   When can one rigorously write $v=v(x)$? How do you distinguish between magic effects that bypass immunity and those that don't? This question was triggered by .  Basically, it goes like this: If you have a weapon with a certain magical effect, let's say the minor property unbreakable (can not be broken, requires special means) or temperate (no harm in temperatures between -20 to 120 F), it can bypass the immunity of something like a Werewolf because it is magical instead of mundane. This goes for any minor property because those properties are under the Magic Items portion of the DMG on pg.143.  How does this differ from a pair of gauntlets with the exact same effect being used as an improvised weapon (emphasis added for clarity) from dealing damage in the same manner to the same Werewolf? Or something that's clearly not a weapon like a Shield of Missile Snaring being used to bash an opponent? How can I tell if a laptop's video card can be upgraded? Undefined control sequence and \chapter error I’m getting an “undefined control sequence” error with the following code, using TeXstudio:  \documentclass[11pt]{article} \begin{document} \begin{center} \chapter{CHAPTER 6} \end{center} \end{document} Why should I use a private variable in a property accessor? How do I check if an element is really visible with JavaScript? In JavaScript, how would you check if an element is actually visible?  I don't just mean checking the visibility and display attributes. I mean, checking that the element is not   visibility: hidden or display: none underneath another element scrolled off the edge of the screen   For technical reasons I can't include any scripts. I can however use  as it is on the page already. In Pillow Problems and a Tangled Tale Lewis Carroll presents the following problem:     A bag contains a counter, known to be either white or black. A white counter is put in, the bag is shaken, and a counter is drawn out, which proves to be white. What is now the chance of drawing a white counter?   One can easily reason that in the end, the bag must be in one of three states:   Black remains, new white taken New white remains, original white taken Original white remains, new white taken   In two of those states, a white counter remains to be drawn. Therefore, the chance is 2/3.  How can one arrive to this answer using Bayes' theorem? Is there a specific term for a statement that uses words such that their precise meaning conveys a subtext contrary to the colloquial meaning?  For example, the joke in Men in Black:     Gentlemen, congratulations. You're everything we've come to expect   from years of government training.   where the precise use of the words conveys a sarcastic meaning (ie that the candidates are inept because that's what he's come to expect from government training).  I was going to use "irony" to describe this but then realized this is basically the opposite of irony - the use of words to convey their literal intention, when a nonliteral intention was expected. Prove that if $A$ is an infinite set then $A \times 2$ is equipotent to $A$ I want to prove that if $A$ in an infinite set, then the cartesian product of $A$ with 2 (the set whose only elements are 0 and 1) is equipotent to $A$.  I'm allowed to use Zorn's Lemma, but I can't use anything about cardinal numbers or cardinal arithmetic (since we haven't sotten to that topic in the course).  I read a proof of the fact that if $a$ is an infinite cardinal number, then $a+a=a$, which is something similar to what I want to prove.   Any suggestions will be appreciated :) past simple or present perfect Power supplies are available in a wide range of voltage and current ratings.  If I have a device that has specific voltage and current ratings, how do those relate to the power ratings I need to specify?  What if I don't know the device's specs, but am replacing a previous power supply with particular ratings?  Is it OK to go lower voltage, or should it always be higher?  What about current?  I don't want a 10 A supply to damage my 1 A device. Why does an environment's label have to appear after the caption? \documentclass{article} \usepackage{algorithm}  \begin{document} \setcounter{section}{54} \section{A section with no label} \begin{algorithm} \label{myalg} \caption{An algorithm with the label `myalg'.} \end{algorithm} Reference to myalg: \ref{myalg}. \end{document}   With this code, the reference givens 55 (the section number); if I move the label after the caption command, I get the correct reference (1).  Why should this be the case? I find it confusing: when thinking about an algorithm as an environment, you expect to be able to label it right after you began it. At the very least I would expect some kind of error or "??" to appear, rather than the previous environment or reference-able entity being used. Why is it called a quadratic if it only contains $x^2$, not $x^4$? Should it not be bidratic or didratic etc?