How to switch back to Unity
Reverting back to Unity (Unity 7)
How to visualize the gradient as a one-form? I am reading Sean Carrol's book on General Relativity, and I just finished reading the proof that the gradient is a covariant vector or a one-form, but I am having a difficult time visualizing this. I usually visualize gradients as vector fields while I visualize one-forms with level sets. How to visualize the gradient as a one-form? How do I delete unwanted Apple ID's? I'm going thru a phone swap, and I now realize that I have multiple ID's   "codename" "codename@example.com" "realname@example.com"   I've tested all three, and they all have different login passwords. Not sure how I got here but I'd like to clean the mess up.  Is there any way to delete / remove / erase unwanted ID's so they don't exist anymore?  (Note: their existence creates total havoc with any type of ID edit...)  I see from  that I can't merge the apple ID's, but can I simply delete the unwanted ID's? The highest common factor and the lowest common multiple of two numbers $A$ and $B$ are $12$ and $168$ respectively. Find the possible values of $A$ and $B$ with the exception of $12$ and $168$.   Help me with this equation please. I kind of get it but I don't understand the concept behind it... I do not want to memorize. A nice user suggested  to . The edit, though very good and desirable, was rejected. I do not understand the reasons, and I did not find a way to contact those people who rejected it. What can I do to re-approve the edit? For example, here are top five accounts in each site (this picture was taken when I was in Biology SE).    As you can see, all the badges are DNA, while Academia's badges should be hats, so on.  What do you think? Do airlines have to check the visa status before selling a ticket? Using `M` instead of `k` when rep is >= 1M on https://chat.stackexchange.com/users I read on  ()    Shouldn't it be 1.03m or (if we really follow the conventions of the metric system) 1.03M instead? Twin Paradox, calculating spacetime intervals from both perspectives I've very recently started to try to understand special relativity. I'm want to get a decent understanding of the twin paradox. I'll post what I've done so far and highlight what's gone wrong for me.  The situation is that Alice and Bob both begin at point $x_1$ in Alice's coordinate system $(x,t)$ (we have orientated the axis so that y and z will not matter). Bob then instantaneously moves off with velocity $v$ in the positive $x$ direction. Bob's coordinate system is now $(x',t')=(\gamma (x-vt), \gamma (t-v\frac{x}{c^2}))$. In Alice's coordinates, Bob reaches point $x_2$ after having moved distance $d$, then instantly turns around and travels back to Alice at $-v$. I want to show that Alice's space time interval $\Delta s_A^2$ is greater than Bob's $\Delta s_B^2$, as each path's space time interval is proportional to the proper time passed along that path.    According to Alice: $\Delta s_A^2=-c^2\Delta t^2 + \Delta x_A^2$ where $\Delta t=\dfrac{2d}{v}$ and $\Delta x_A=0$ as she didn't move, but the time elapsed is the time it took bob to travel the distance and then come back. Thus $\Delta s_A^2=-\dfrac{4c^2d^2}{v^2}$  Still according to Alice, Bob's $\Delta s_B^2=-c^2\Delta t^2+\Delta x_B^2$, where the change in time is the same, but now bob has moved distance $d$ twice, thus $\Delta x_B=2d$ ($\Delta x$ refers to total distance traveled rather than displacement which is 0 in this case). Now we have $\Delta s_B^2=-\dfrac{4c^2d^2}{v^2}+4d^2$.     The |size| of Bob's spacetime interval now definitely smaller than Alice's, and this would be all OK, except that when I do the calculations in Bob's frame, they don't agree. This is contradictory to the fact that $\Delta s^2$ is conserved under Lorentz transformations.    According to Bob: $\Delta {s'}_A^2=-c^2\Delta t'^2 + \Delta {x'}_A^2$ where $\Delta t'=\dfrac{2d'}{v}$. I'm not sure if I'm right in saying that $v$, the relative velocity of the two frames, is the only velocity upon which both Alice and Bob will agree on, other than the speed of light. Anyway, $d=\gamma d'$ as lengths contract by $\gamma$ i.e. $d'$ is smaller than $d$ by factor $\gamma$. Also, $\Delta {x'}_A=2d'$, so we have $\Delta {s'}_A^2=-\dfrac{4c^2d'^2}{v^2}+4d'^2=-\dfrac{4c^2d^2}{\gamma ^2 v^2}+\dfrac{4d^2}{\gamma ^2}=4d^2\dfrac{v^2-c^2}{\gamma^2 v^2}\neq \Delta s_A^2$  Although I haven't included them here, my calculations for $\Delta {s'}_B^2$ and $\Delta s_B^2$ agree. Sorry for the ultra-long post, but any help would we well appreciated! How can I search in another language? "Distribution not supported" when trying to install Intel Graphics Installer in Ubuntu Keyboard not working after suspend to ram (sleep) How to measure GPU usage? Is it possible that by mutation a human could see infrared or other 'colours'? Incoming light reacts with the several types of cone cells in the eye. In humans, there are three types of cones sensitive to three different spectra, resulting in trichromatic color vision. Each individual cone contains pigments composed of opsin apoprotein, which is covalently linked to either 11-cis-hydroretinal or more rarely 11-cis-dehydroretina.  The opsins (photopigments) present in the L and M cones are encoded on the X chromosome.  A very small percentage of women may have an extra type of color receptor because they have different alleles for the gene for the L opsin on each X chromosome. X chromosome inactivation means that only one opsin is expressed in each cone cell, and some women may therefore show a degree of tetrachromatic color vision. Tetrachromacy is also demonstrated among several species of birds, fish, amphibians, reptiles and insects.  Humans cannot see ultraviolet light directly because the lens of the eye blocks most light in the wavelength range of 300–400 nm.  So, besides above mentioned exceptions, is it possible due to mutations or genetic engineering to see UV or infrared. What kind of mutations should be made or are the changes to be made in the opsines to big to create 'alien' eyes? How do I know which edition of Dungeons and Dragons (D&D) the books I'm looking at are for? , each of which has loads of rules, source, and splat books.  But how do I know which edition the books I'm looking at are for?  A lot of the time the ones in hobby stores are wrapped up, or you'll buy from elsewhere that doesn't say in the description, and the covers just say D&amp;D.  Is there a way to identify which edition of D&amp;D the books I'm buying are for, either from the cover or some description in the book?  For the purposes of this question I'm asking about core rule books only. Eg. PHB, MM, and DMG. Hopefully the answer to this can be carried over to other books. Prove $\sum_{d \leq x} \mu(d)\left\lfloor \frac xd \right\rfloor = 1 $ Programming pattern to generate VGA signal with micro-controller? I want to generate VGA signal with micro-controller (like TI Tiva ARM which runs at 90/120Mhz speed).  I'm not sure how to make accurate timings with micro-controller. Which programming pattern I need to use?  Do I need any inline assembler code? How to wisely use interrupts?  Would be great if anybody show some pseudo code how to generate VGA signal.  I successfully generated VGA signal with FPGA. But I just can't figure out how to do it with MCU. Refused UK Visit visa for lack of sufficient funds - What more can I do? I'm writing a mildly dystopian scenario where competition is taken to vicious extremes, and I was wondering if there was a word that I could use (specifically a noun that was a synonym of "competition") to describe the sort of vicious, cutthroat ... _________ (the word that I want) that was going on.  I'm specifically looking for a word that doesn't refer to a one-off event (like "contest" or "tug-of-war"), but rather is more about the state of things. "Activity" tab for the Community user's network profile sometimes leads to a server error If you finish a world without fighting the Broodal, can you still fight them later? I've just seen that in a speedrun: If you collect enough moons to move on to the next world, and cash them in at the ship, the broodal ship just leaves.  Does that mean that the Broodal fight wont happen, so you missed the triple moon it gives? Or if you go to the arena, even without the ship you will have the fight happen?  Clip where this happens :