Simple/elementary explanation for $\mathbf{3} \otimes \mathbf{\bar{3}} = \mathbf{8} \oplus \mathbf{1}$?
$SU(3)$ irreducible representations with tensor method
What should children call a step-grandparent? My mom remarried to a great guy who has been absolutely wonderful to my kids. Ever since my mom and he were dating my kids have called him Mr. Joe. Lately my mom has been bugging me to get the kids to call him Grampa.  I agree that he deserves more respect than Mr.  The problem is that he isn't their real grampa so I don't really want them calling him that. Does anybody have any suggestions on what we should have the kids call him? They are 9 and 12 years old.  EDIT: I appreciate the comments regarding different variations of grampa and hearing how others have no issues with calling their step-granddad grampa. But I was hoping maybe someone has some suggestions for names that show the importance of the step-granddad without calling him some variation of the grampa name but still has a feeling of warmth and family in the name (I hope that makes sense). When editor does not respond to the request for withdrawal I submitted a manuscript to a journal, but due to some issue with that I asked the Editor-in-Chief and Associate Editor to withdraw the manuscript from further consideration. Despite several emails, they are not still responding me nor ended the process.  Now, what should I do? Can I submit the manuscript to another journal? Keep sharp edges when using subdivision surface How can I use the subdivision surface modifier to create more high poly meshes without ruining sharp edges? For example this mesh  The sphere should be subdivided but the rim of the cylinder should stay the same because it is a sharp edge. This is the result of the subdevision  How can I get sharp edges to not be affected by the sub surf mod? How to make a field NULL using MySQL GUI Tools? I'm using the MySQL Query Browser (part of the ) and need to change a field to NULL, but I can't figure out how to do it - if I delete the value it tries to update it to ''. Typing "NULL" makes it try to update to 'NULL' (a string).  I know I could just write a query to do it, but that defeats the entire purpose of the tool, no? A person who cannot understand that there may be other perspectives Formula for completing the square? My math teacher said that this was the formula for completing the square.  Original function: $$ax^2 + bx + c$$ Completed square: $$a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c$$  However, using this formula I'm not getting the same answers that I would get just by determining the stuff myself.  Is this correct? Correct usage of "see" vs. "watch" Short story: people rediscover how to do math and want to put people in bombs Many years ago I remember reading a short story about an advanced civilization at war that rediscovers how to do math mentally. The reason I remember it is the guy tells it to government officials who are delighted because it means they can put people inside bombs or missiles instead of computers, which were thought to be more valuable.  I believe this is one of those stories I read in a science fiction anthology my father brought home on one of his business trips in the sixties or seventies. When $d$ is in $\mathbb{Z}$, $\mathbb{Z}[\sqrt{d}] = \{a_0 + a_1 \sqrt{d} \mid a_0, a_1 \in \mathbb{Z}\}$  However, when $d$ is in $\mathbb{Q} \setminus \mathbb{Z}$, I cannot prove it.  Of course, when I see an example, I can understand that this is not a finitely generated abelian group, but I want to know the proof in general. How do I halt an android phone using adb command? I can reboot, but I don't know how to halt.  I tried shutdown -k wait command, but it did not work. I installed an Ubuntu alpha or beta and have been keeping it up to date.   So, Should I presume that my system has automatically updated to the next alpha or beta, and will be running the official release when it's finished? Migrated questions don't have a user action label in the question list I would like to solve the beginner's standard exercise which claims that a point of $\mathrm{Spec} \ R$ is closed iff it is a maximal ideal.  The reverse implication is easy. The direct one seems much subtler. If $P = V(I)$ then, since there exist a maximal ideal $M$ containing $P$, it would follow that $\{P, M\} \subseteq V(I) = P$, whence $P=M$. In order to show the existence of $M$ I applied Zorn's lemma to the set of ideals containing $P$.  The question is the following: is there any other proof of the above implication that does not (indirectly) use the axiom of choice? My concern is that I may be using a cannon to shoot a fly. Also, it would be the first time that I see closedness of points to require the axiom of choice. If the axiom is indeed needed in general, are there "nice" classes of rings for which we could get away without it? What software can I use to view epub documents? What software can I use to view .epub documents? Where does the energy for osmosis come from? A thought experiment:     A U-shaped tube with semi-permeable membrane at the base. The tube is completely thermally isolated from its surroundings. The liquid (solvent) is at some temperature $T$. When solute is added to one side of the tube the fluid level on that side rises until the increased pressure due to gravity ($\rho gh$) equals the osmotic pressure.   If I understand correctly, osmotic pressure could be used to do work (i.e. lift a weight as the solvent level in one side of U-shaped tube increases). Where does this energy come from? I guess it must be from the internal energy of the fluid.  At this point has the temperature of the fluid decreased?? If so, has it decreased by exactly the work done against gravity divided by the specific heat of the fluid? What does the quantitative treatment of this problem look like? Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. How can I completely disable internal Mic and Webcam I want to disable the internal Microphone on my laptop and the internal Webcam completely that they're no longer selectable, and no longer available?  I will only use my external Webcam, and my external (USB) headset.  The internal audio should still be available for playback. Editable fields in views The following identity is true for any given $x \in [-1,1]$: $$\arcsin(x) + \arccos(x) = \frac{\pi}{2}$$  But I don't know how to explain it. I understand that the derivative of the equation is a truth clause, but why would the following be true, intuitively?  $$\int^{x}_{C1}\frac{1\cdot dx}{\sqrt{1-x^{2}}} + \int^{x}_{C2}\frac{-1 \cdot dx}{\sqrt{1-x^{2}}} =\\  \arcsin(x) - \arcsin(C1) + \arccos(x) - \arccos(C2) = 0 \\  \text{while } \arcsin(C1) + \arccos(C2) = \frac{\pi}{2}$$  I can't find the right words to explain why this is true?    Edit #1 (25 Jan, 20:10 UTC): The following is a truth clause: $$ \begin{array}{ll} \frac{d}{dx}(\arcsin(x) + \arccos(x)) = \frac{d}{dx}\frac{\pi}{2} \\ \\ \frac{1}{\sqrt{1-x^{2}}} + \frac{-1}{\sqrt{1-x^{2}}} = 0 \end{array} $$  By integrating the last equation, using the limits $k$ (a constant) and $x$ (variable), I get the following:  $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = m  \text{ (m is a constant)}\\ \\ \arcsin(x) + \arccos(x) = m + \arcsin(k) + \arccos(k) \\ \\ \text{Assuming that } A = m + \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$ Using Calculus, why is that true for every $x \in [-1,1]$?    Edit #2:  A big mistake of mine was to think that $\int^x_k0 = m \text{ (m is const.)}$, but that isn't true for definite integrals.  Thus the equations from "Edit #1" should be as follows: $$ \begin{array}{ll} \int^x_k\frac{1}{\sqrt{1-x^{2}}}dx + \int^x_k\frac{-1}{\sqrt{1-x^{2}}}dx = \int^x_k0 \\ \\ \arcsin(x) - \arcsin(k) + \arccos(x) - \arccos(k) = 0\\ \\ \arcsin(x) + \arccos(x) = \arcsin(k) + \arccos(k) \\ \\ A = \arcsin(k) + \arccos(k) = \frac{\pi}{2} \text{ ,for } x \in [-1,1] \end{array} $$ How does SO's (or other SE sites) versioning work of questions and answers? Do they use some standard piece of code or is it something homebrew?  With versioning I meant if for example a question is edited you'll see all the changes (a diff).