After defining certain colors and modifying them as the following code indicates, I then would like to determine the equivalent RGB definitions of the modified colors. For example, \documentclass{article} \usepackage{tikz} \newcommand{\mycbox}[1]{\tikz{\path[draw=#1,fill=#1] (0,0) rectangle (3cm,3cm);}}  \definecolor{red}{rgb}{1, 0, 0} \definecolor{blue}{rgb}{0, 0, 1} \definecolor{burntorange}{rgb}{0.8, 0.33, 0.0}  \begin{document}     \mycbox{red}      \mycbox{blue}     \mycbox{burntorange}          \vskip 30pt  \textbf{\LARGE{RGB Definitions of These Colors? ---}}  \vskip 15pt      \mycbox{red!80}      \mycbox{blue!60}     \mycbox{burntorange!65}     \mycbox{burntorange!55!red!80} \end{document}   How may I determine the exact RGB definitions of the four colors in the second row? Thank you.
How to exactly convert an arithmetic rgb value (e.g 0.55,0.25,0.1) to the corresponding 8bit RGB value (e.g. 123,90,40)?  My more specific questions are:  1.) What is the exact conversion formula and macro?  2.) Is there some built in command or macro for converting it?
How to use combinatorial proof instead of induction? Let k ≥ 1 be an integer and consider a sequence $n_1, n_2, . . . , n_k$ of positive integers. Use a combinatorial proof to show that  $\binom{n_1}{2}$ + $\binom{n_2}{2}$ +...+ $\binom{n_k}{2}$ $\le$ $\binom{n_1 + n_2 + ...+ n_k}{2}$ Im confused about fire. The way I see it :   Heat creates (kinetic) energy in mass and this creates stronger vibrations of atoms. When those vibrations are strong enough the electrons interact stronger due to electromagnetic forces. This causes the electrons to fly away. This is the creation of a plasma and the direction of most electrons is the direction of the flame. the shape and heat of the flame depends mainly on the amount of electrons and their velocity ( apart from the source of heat and the considered gas and solid material) the electrons move freely ( but not random ) in the flame. there is a chemical reaction going on that usually produces smoke or such. the entropy increases in all the above 6 steps and the process is repeated. the electromagnetic energy is converted into kinetic energy of the electron and conductivity in the plasma (flame). the moving electrons create an electromagnetic field.   That is more or less how I see it. What steps are ok ?  And the main question , why is there light involved ?  I mean why cant we just have kinetic energy of the electrons and a changing magnetic field ? Why are there photons ? If the electrons are in a high energy state why cant they just move faster instead of emitting a photon ???  Why is the light usually visible ? is that because of our structure of air ?  If the light is an electromagnetic wave why does the flame go up ? does that imply all electromagnetic waves go up ?? But electromagnetic waves are not influanced by gravity or buoyancy are they ??  What is the final destiny of the electrons and photons if the heat is gone ??  I am puzzled by this. How can a credit card company make any money off me? I have a no-fee card and pay my balance on time How to create Mist in Blender I already tried several ways to create Mist in Blender, but it never worked 100%. I use Blender 2.78 and I want to create Mist like in this scene here.  What are the exact steps?     Here is what I have tried so far:  I already tried this:  but it simply did not work .. I dont know why.  (EDIT: I managed to fix it: Check out this graphic:   Until now I already used objects successfully as Mist-creators (Volume Scatter under Volume Material). The bad thing about using Objects as Mist-creator is that they dont have an endless mist range + the mist can not be bright. Normally it appears grayish result is bad..  Check it out  Please help Gauge fields and strings: Loop equations Single service account running all SQL Servers Is there any negative performance impact for using the same service account and agent account to run SQL Server and SQL Agent respectively for all SQL server instances running at a small company of say 35 servers? The biggest database is 0.5 GB. any suggestions are welcome. Thank you. Can XeLaTeX generate bold and italic font styles for typefaces that lack them? I use XeLaTeX in order to have easy access to fonts installed on my operating system, and Chinese in particular. Some of the Latin typefaces most pleasing to my eye do not have bold or italic font styles, and I wonder if XeLaTeX can remedy that.  I'm aware of the FakeBold and FakeSlant transformations within the fontspec package, but I'd be glad to know of a more attractive process, if there is one. (The specification, v2.1f, says, "Please don’t overuse these features; they are not a good alternative to having the real shapes.") How do we control web page caching, across all browsers? Vectors, structs and std::find Suppose $f$ is an isometric (i.e., distance preserving) function on $\mathbb{E}^2$ such that $f(0,0) = (0,0)$.  Then I want to show that $f$ is necessarily linear.  Now $f$ is linear iff $f$ is both additive and homogenous.  The following is an attempted proof for the homogeneity of f (missing the last step); still more, I have no idea how to argue for the additivity of $f$.  Any ideas?  Let $x \in \mathbb{E}^2$ and $\alpha \in \mathbb{R}$   We know that $\forall x \in \mathbb{E}^2$, $\Vert x - 0 \Vert = \Vert f(x) - f(0)\Vert = \Vert f(x) - 0 \Vert$ so that $\Vert x\Vert = \Vert f(x)\Vert$.  From this we immediately have the following facts:  $\Vert x \Vert = \Vert f(x) \Vert$  $\Vert \alpha x \Vert = \Vert f(\alpha x) \Vert$  We can then argue that since $\Vert \alpha x\Vert = |\alpha| \Vert x \Vert = |\alpha| \Vert f(x) \Vert = \Vert \alpha f(x) \Vert$, we also have that $\Vert f(\alpha x) \Vert = \Vert \alpha f(x)\Vert$.  Finally, we have that  $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$   iff $ \Vert (\alpha - 1)x \Vert = \Vert (\alpha - 1) f(x) \Vert$   iff $|\alpha - 1| \Vert x \Vert = |\alpha - 1| \Vert f(x) \Vert$  Since the last of these statements is in fact true, we now have $\Vert \alpha x - x \Vert = \Vert \alpha f(x) - f(x) \Vert$ as desired.  Now at this point it seems like I have all of the facts required to assert that $f(\alpha x) = \alpha f(x)$, but I can't figure out how to formally state why without illegally appealing to visual intuition.  Any ideas? How can I create a product key for my C# application? How can I create a product key for my C# Application? I need to create a product (or license) key that I update annually.  Additionally I need to create one for trial versions.  Related: In my textbook, a relationship is plotted between the resistance of the material and the temperature, and the temperature coefficient is defined as the slope of that graph divided by an arbitrary resistance $R_1$ on the graph. Does that mean that the coefficient varies depending on the temperature I choose to divide the slope by?  If so, how does it stand as a valid reference to the material’s resistance growth rate with temperature?  If not, then where did I go wrong with this train of thought? Are all integers fractions? How do I play a YouTube video in the background on my iPad? I like to 'listen' to YouTube video in the background while I am surfing on my desktop. However, I cannot do this on the iPad as, as soon as I switch from YouTube video to the browser, the video pauses. Is there a solution for this either in the form of an app or a website? What's the quickest Schengen consulate to issue a visa in London? What's the quickest Schengen consulate to issue a Visa in London? Basically I want to travel as fast as possible to France, and since any Schengen visa will work, I want to go to the consulate with the shortest processing time. Invoking makeindex for nomencl I have spent many many hours failing to implement nomencl for my PhD thesis. The problem seems to be with invoking makeindex and creating the .nls file  I keep reading that I must invoke makeindex or write a Unix command:  makeindex filename.nlo  -s nomencl.ist -o filename.nls   But I don't know how to do this. I would be grateful for some instructions i.e. where do I type the command?    I am using TeXShop 2.43. How to "flatten" a multi-dimensional array to simple one in PHP? It's probably beginner question but I'm going through documentation for longer time already and I can't find any solution. I thought I could use implode for each dimension and then put those strings back together with str_split to make new simple array. However I never know if the join pattern isn't also in values and so after doing str_split my original values could break.  Is there something like combine($array1, $array2) for arrays inside of multi-dimensional array? Running two MySQL servers that use the same data directory Booking a flight without passport info I have been looking to book a bundle (hotel and flight) with Expedia to Puerto Vallarta MX. When I get to the booking section it does not ask for my passport number. Do they ask for my passport after I book? Or later when the date gets near? My passport is currently being renewed and I want to book my hotel and flight before they sell out but I am afraid it'll ask me for my passport number. Can anyone inform me about this situation? What's wrong with this argument that $[0,1]$ is countable?