I am busy working on a 3D $n$-body galaxy simulator and I am having some difficulty wrapping my head around determining the initial velocity components of each particle to ensure a circular orbit around a central mass.  I know that the magnitude of the velocity to ensure a circular orbit is calculated from: $$ v = \sqrt{\frac{GM}{r}} $$  How then can I determine what the individual velocity components in 3D will be for the particle?
I'd like to write an N-body simulation in which I collide two disk galaxies. To give you an idea of the accuracy I'm trying to achieve, I'm aiming to make this my screensaver at 30fps on my work desktop. Think Euler's method, large epsilons in denominators to prevent blow up, all that lazy stuff.   One place I don't want to be lazy, however, is in the initial construction of the galaxies I'm trying to collide. I want to design a galaxy that will at least sort of hold together by its own gravity until it smash it into another one.   How might I go about selecting the orbital velocities for each of the bodies composing the galaxy?
What is the net charge of the Earth? How can I force a line break in the table of contents? How does "Reputation" work? I'm trying to work out this shoulder deformation. I can't figure it out.  Is there anything wrong in my armature? Should i add bones and keep more control of my mesh or use shape keys? Here is my file blend. What is the distribution of primes modulo $n$? Remove GRUB and keep a Linux system Why should a CTE start with a semi-colon? I was just looking at a post on  where Aaron Bertrand proposes using a CTE instead of a numbers table, which is an elegant way of performing the task at hand.  My question is, why does the first line of the CTE begin with a semi-colon?    ;WITH n AS (SELECT TOP (10000) n FROM    (SELECT n = ROW_NUMBER() OVER     (ORDER BY s1.[object_id])     FROM sys.all_objects AS s1     CROSS JOIN sys.all_objects AS s2   ) AS x ORDER BY n ) SELECT n FROM n ORDER BY n; -- look ma, no gaps!   Is this to ensure the WITH statement does not get parsed into a previous SELECT or something?  I see nothing in SQL Server 2005 BOL about using a semi-colon prior to the WITH. How do you get JavaScript/jQuery Intellisense Working in Visual Studio 2008? I thought jQuery Intellisense was supposed to be improved with SP1. I even downloaded an annotated version of jQuery 1.2.6, but intellisense will not work in a separate jscript file. I have the jQuery library referenced first on my web page in the &lt;head> tag. Am I doing anything wrong? Simulating the Windows key on an old keyboard How to reconcile the cashless human/Federation society with human traders and merchants From a variety of sources, including , we know that the Federation (or at least the humans) don't use money. This raises the problem that humans frequently visit worlds where there is currency, and, in the case of DS9, operate a station that has currency frequently in use (for example, Federation personnel frequently attend Quark's bar). Additionally, we see human traders.   My question is, how do humans get "money" (credits, latinum, etc.)? How would a human even get the resources to purchase their own merchant ship or the resources to fill its cargo holds with merchandise? If Federation personnel can simply be given money for the asking, why haven't they poured so much money down Quark's throat that he chokes on it? How do I change my timezone to UTC/GMT? Currently when I execute the date command from bash I get the following displayed  Thu May 17 12:21:00 IST 2012  Below is my timezone config based on executing dpkg-reconfigure tzdata     Current default time zone: 'Europe/Dublin'   Local time is now: Thu May 17 12:32:52 IST 2012.   Universal Time is now: Thu May 17 11:32:52 UTC 2012.     Any idea how I change from IST to UTC/GMT? Linux ACL behaviour: settings the mask changes 'ls' output, but not actual permissions It might be the time of night, but this is puzzling me. Picture the following.  [root@node1 acltest]# getfacl foo/ # file: foo # owner: root # group: testuser user::rwx group::r-x other::---  [root@node1 acltest]# ls -la . total 24 drwxr-xr-x  3 root root     4096 Feb  9 21:53 . drwxr-xr-x 25 root root     4096 Feb  9 21:54 .. drwxr-x---  2 root testuser 4096 Feb  9 21:53 foo [root@node1 acltest]# setfacl -m m::rwx foo [root@node1 acltest]# getfacl foo/ # file: foo # owner: root # group: testuser user::rwx group::r-x mask::rwx other::---  [root@node1 acltest]# ls -la . total 24 drwxr-xr-x   3 root root     4096 Feb  9 21:53 . drwxr-xr-x  25 root root     4096 Feb  9 21:54 .. drwxrwx---+  2 root testuser 4096 Feb  9 21:53 foo [root@node1 acltest]# su - testuser [testuser@node1 ~]$ cd /acltest/foo/ [testuser@node1 foo]$ ls -la . total 16 drwxrwx---+ 2 root testuser 4096 Feb  9 21:53 . drwxr-xr-x  3 root root     4096 Feb  9 21:53 .. [testuser@node1 foo]$ touch bar touch: cannot touch `bar': Permission denied   In words: I create a directory foo, with mode 0750, root as owner and testuser as the group. (testuser is the private group of testuser, but that doesn't matter.)  The getfacl command correctly shows no ACL's on this directory and there is not mask yet. The mask will be set accordingly to the group permissions if I would add a named group or user now.  If I explicitly set the mask to rwx now, the group permissions as shown by ls change too. I know it happens the other way around (the mask changes when the group permissions change), but this seems puzzling.  The more puzzling because the getfacl output does not show the group permissions as rwx but - as said - ls does.  Which one is right? Apparently, the output of getfacl is right, because testuser cannot write to foo. As expected, by the way, because I did not grant the testuser group any permissions to do so.  It goes on. I cannot allow the testuser group write permissions on foo by just using chmod. I have to explicitly set an ACL using setfacl -m g:testuser:rwx foo to allow testuser to finally touch foo/bar.   Can someone explain the rationale behind the difference in the outputs of getfacl and ls? I know it can be tricky to have normal permissions go together with ACL's, but this seems plain wrong. (Though I expect to be missing something glaringly obvious ;))  I have already seen Change \textwidth after first page in scrlttr2 Why do universities have to spend money on journals? Obviously this is a question in the light of the recent Elsevier boycott. Currently we do have an arXiv, maintained by academia and where researchers regularly upload parts of their work. In such a case,   Why do universities spend lots of money to publish in third-party journals?   The question especially applies to journals that operate with a rigorous profit motive. The subscription is very high, so wouldn't publishing in such journals affect the paper's citation count and deter the spread of knowledge about the work within academic circles?   Why should not universities collaborate to create free, open access, peer-reviewed journals?   Moreover, given the need to conserve paper, why should journals spend on printing research papers? Wouldn't an online version suffice, as most people use only local computer printouts anyway? In other words, why can't we have a Wikipedia-like system of sharing research knowledge, having properly established standards for such journals? Not getting email alerts on answers on Stack Overflow Not getting email alerts on answers on my questions on stackoverflow.com   Used to get it earlier (>1 month back), but not getting alerts in recent times Email field in profile is filled  In prefs, "I'd like email notifications of activity on questions I've asked" is ticked.   What is missing?  Is it possible that signing up on another Stack Exchange community (say, Server Fault), on which by default, "I'd like email notifications of activity on questions I've asked" is unchecked affects same preference on stackoverflow.com? If so, it is a bug. Possible Duplicate:                If in a multiple linear regression (enter method) the general model isn't significant (F>.05) but one of the predictors is significant (β&lt;.05), should I consider it as a significant result? Finding real roots of negative numbers (for example, $\sqrt[3]{-8}$) How can I tell what year my bike was made? This is a canonical question that will hopefully encompass all of the questions we get asking us to determine what year a bicycle was manufactured (how old is my bike, how old is my frame, etc.). Each "answer" should address a different way to determine the manufacturing year of a bicycle. To some extent, these will also help you narrow down the model as well as it will tell you what distinguishing features to look at.  The span of answers would include things such as:      Skiptooth cranks Cartridge bottom brackets (BB30/90 etc) Elliptical chainring Quick releases versus contemporary through-axles How the steerer attaches to the fork (quill vs. threadless)  Braking system (especially with mountain bikes: v-brake, direct pull, etc.)   Shifting ramps and pins on cassettes/sprockets and chainrings (versus flat sprockets) Frame material (e.g., shift from tubular steel to use of CF and hydroformed aluminium)  Frame tubing type (e.g. Reynolds 531 tubing introduced around 1934, still used today; Reynolds 501 tubing introduced around 1983). So a frame with an original 501 decal can't be from the 70', 531 decal doesn't help as much because of the range of years produced. Also, see Decals.   *Decals: style/design, amount, placement, color, etc. can help not only determine models, but also year of manufacture. Many changed decal design for the same model bike often yearly, with special/limited editions even more specific. This applies only if decals are original or have been replaced with the identical design. Tubing decals (Reynolds, Columbus, etc.) also changed designs for same tubing made in different years, although the changes aren't made as often, and manufacturers didn't always put tubing Brand decals on every model. Earlier bikes tended to have less fancy, less colorful, less quantity of decals.   Headbadges: most early bikes had actual "badges", often quite detailed and fancy, made of metal, then plastic, and finally just using decals on the headtube. Most bikes lost their metal badges in the 60's and early 70's, although some brands still have actual badges to this day, but in general a real badge indicates an earlier model bike. Dropout style (including the later inclusion of lawyers lips) Presence or absence of CPSC reflectors (USA only) Manufacturing method (shift from lugs to TIG welding, for example)  Number of gears (shift from 2x5 ten-speed to 3x8, 2x9, 1x11, etc.)  Inspection of lug design: pantographs, cut-out, stampings, custom brazing/filing, chrome, fork crow n sweep, etc. Frame/Fork Braze-ons. Race bikes tended have less braze-ons the earlier the year of manufacture. Most 60's racers and earlier had few or none with cable guides/stops, shifters, bidons, etc. being clamp-on. By the 80's if it was clamp-on, it was a sign of very cheap models and most frames had braze-ons for all cables, with multiple bottle and accessory bosses depending on style and use.   and so forth. Please edit this question if you have an answer that isn't in the index above.   A couple of caveats:   A previous owner could have replaced or upgraded components on a bicycle, so this guide only applies to original equipment There is a delay between the introduction of a technology (such as indexed shifting) and when manufacturers start selling products that use it. Low-end manufacturers may continue using technology that is severely outdated (such as one-piece cranks, freewheels, or quill stems) because it is cheaper to do so. A number of bikes are built in a Retro style, but with modern components.  Dutch bikes, trendy coffee cruisers, and beach cruisers may be modern but appear to be 1950s styling.  These tend to have modern rims and brakes as a giveaway.     Note that bicycles.stackexchange does not do valuations of bicycles. How much your bicycle is worth depends on the location of the buyer and seller, how much the buyer wants the bike, etc. etc. etc. I am running 12.04 LTS. It's a pretty default installation and is the only OS on this machine.  I have two users one has administrator rights the other does not. I accidentally removed administrator rights from the "admin" user.  I tried to  but that requires reaching GRUB during boot but I cannot get there because my /etc/default/grub has GRUB_HIDDEN_TIMEOUT=0 and changing that to a positive integer requires admin privileges to save the changes and run sudo update-grub.  Should I boot from LiveCD? The  discouraged my a bit. Right now all I have handy is an install DVD from 11.10, but that did not help me much.  Not sure what to do now.  Any suggestions?  NOTE: 12.04 does not have an admin group and I have not enabled root Why are we scared of singularities? I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner so much hate. To me, it seems like negative or complex numbers: we used to hate these things but now they are more generally accepted. Why can't it be that nature just has singularities? We might not get it, we might not like it, but it may be so.