Consider $n$ points chosen uniformly and independently on the surface of a sphere with circumference  $1$ at the equator. What is the expected minimum (great-circle) distance between any pair of points?   If we considered a circle instead the answer would be $1/n^2$.
What is the expected minimum euclidean distance between $N$ points uniformly and independently chosen on the shell(sphere) of 3-D ball of radius $R$?   Note that the expected minimum distance might be difficult to compute, so a good lower bound is also fine.   My approach  I take the approach similar to this   .   Suppose we put $N$ circles of radius $r$ uniformly on the surface of the ball. Therefore we could compute the expected minimum distance by the following expression \begin{align} E[D]=\int_0^\infty P(D&gt;t) dt. \end{align}  So, it remains to compute  $P(D&gt;t)$. Actually, since we are after a lower bound on  $E[D]$ it is enought to give a lower bound on $P(D&gt;t)$.  Let $S_i$ be the event that the pair of circles does not intersect which should be given by \begin{align} P(S_i) =  \frac{ Surf(R)-Surf(r)}{Surf(R)}= 1-\frac{  \pi r^2}{ 4\pi R^2}=1-\frac{ r^2}{ 4 R^2} \end{align}  Now the probability that euclidean minimum distance $D$ between $N$ balls is bigger than $t$ should be similar to  \begin{align} P[ D \ge t] \stackrel{?}{=}  P[ \cap S_i] \end{align} However, I think we somehow need to converte from the geodesic distance to euclidean distance but I am not very sure how to do that?
How can I programmatically delete a line from a Word document using c#? What is the net charge of the Earth? I know that to better balance paragon and epic level players against similar level monsters, WotC changed the math used for monsters for MM3, and I believe that they changed some of the types of powers they gave monsters.   If I want to update monsters from MM1 or MM2 to be in line with MM3 era monsters, what changes should I make? Why is len(Posts.body) incorrect on answers in data.SE? I was trying to find a longest answer on a site, and ran into a weird discrepancy:  shows that  has a length of 36388 (length in the query is defined as LEN(Posts.Body)) Yet, the post clearly only has ~26000 characters, as measured 2 ways:  Copy/pasting full text of the answer in my text editor gives me 26040 characters (the source of the answer with all the markdown, not the rendered text which is even smaller).  Trying to post a new answer that consists of this answer's text twice gives me an error &quot;Body is limited to 30000 characters; you entered 52258&quot; - meaning that the length is under 26100 (not 100% sure where the difference in counts comes from as far as two methods)   Another example from same query:  shows len(body) of 33738 whereas the text of the answer has only 29000 characters. This clearly isn't showing rendered characters, as the rendered character count is LOWER than actual answer text when editing (e.g. links add characters, as do images). How can this data.SE discrepancy be explained? Java generics super keyword Setting up LVM Snapshot as a backup/restore point in ubuntu Setting up LVM Snapshot as a backup/restore point in ubuntu. Why? Well, as I see it, with all the great usefulness of Linux, its biggest drawback is with its mixed up way of installing apps. All the dlls go to one directory. All the configuration files to another. And the binaries are also put in a bunch. Yes, Windows works the same way more or less, but with the nature of system/server software installed on Linux, being that most of the software installed is only about 95% mature at any given time, messing up your system twice a month is very probable.. Of course, we can backup the system before and after each installation, but that takes a million years. System backups (as oppose to just data/files backup) requires you to turn off you computer and use some form of bootable live CD and a backup media. And, if the drive is more than 15% full, it might take a long time to do the backup, even with modern hardware. So, how can this be done? When using an Int- or FloatProperty, is there any way of changing their min and max values dynamically?  I'm currently loading images on the fly and would like to limit a frame property to run from 0 to len(loaded_images) every time I call the loading operator. However, the property seems to be read-only and re-assigning gives the following error:  bpy.context.scene.project_props.bl_rna.properties['myIntProp'] = bpy.props.IntProperty(...) TypeError: bpy_prop_collection[key] = value: invalid, expected a StructRNA type or None, not a tuple   I understand the bpy.props constructor uses an intermediary tuple, but I'm not sure how to proceed. Is there any way around this? I have desktop edition of Ubuntu.  I like the Terminal so that that I prefer to shutdown my computer with the shutdown command.  However when I type shutdown now it prompts me to enter my password.Is there any way I can shutdown my computer using this command without entering my password? Recording claimable expenses in GnuCash Is there a page listing all fields? I've seen them maneuver. They definitely can go backwards. How fast can they go in reverse? Is warp speed possible, or is reverse limited to impulse power? If it is limited to impulse power, can they travel backwards under impulse power as fast as they travel forwards under impulse power? How can I set custom key bindings and key macros on Mac OS X? I recently upgraded to Snow Leopard, from Leopard and I was using an awesome app called  that let you remap your keys, set scripts/macros and even had repeat functionality.   However, this app doesn't seem to be in active development anymore, and 10.6 broke the ability to bind certain modifier keys (Shift) with other keys.   I totally miss things like Shift+Insert for paste (Linux/Windows), setting Ctrl+Shift+Esc for Activity Monitor, and Ctrl+Alt+Del for the login window.   So, how can I achieve this instead? Let us say you are in an inertial reference frame with a circular planar disk. If you take your meter measuring rods (or perhaps tape measure) you can find the diameter and circumference of the disk. If you divide the circumference by the diameter, you will get exactly $\pi$. Now you start rotating the disk. A book I have claims that now the ratio will be different (lets call it $\pi_\circ$ to avoid ambiguity.) This has caused to reevaluate my understanding of special relativity (this is a set up to general relativity, but the problem itself only requires the special theory.) My problem is that, sure, the measuring rods (or tape) would contract on the circumference, but wouldn't the circumference contract a similar amount, therefore giving a measurement of $\pi$ in every reference frame according to your own measuring rods. Of course, this reasoning goes on to make me question how I understood all of special relativity up this point. Why does the find command blow up in /run/? I used the following command to check python versions as given by . It does work except for one error, shown below. Can any one explain why?  $ sudo find / -type f -executable -iname 'python*' -exec file -i '{}' \; \     | awk -F: '/x-executable; charset=binary/ {print $1}' \     | xargs readlink -f | sort -u | xargs -I % sh -c 'echo -n "%: "; % -V'   Error:  find: ‘/run/user/1000/gvfs’: Permission denied I'm using biblatex for my references.  Normally, if I use \cite{source1, source2, source3}, I get a citation like  [3, 45, 98]   If I now want to give a page number for one of them, it looks confusing imho, as I should get (did not try it out up to now, honestly)   [3, 45, 78, p. 23f, 98]   So imho it would make sense to change one of the separator from a comma to e. g. a semicolon like:  [3; 45; 78, p. 23f; 98]   Question 1: Does that make sense, or are there other conventions for citing sources with and without page numbers together?  (Up to now I collected the citations without page number in one cite command and added singular \cite[][p. xy]{} commands for those with page numbers., so I'd get:   [3, 45, 98][78, p. 23f]   )  the \cites[]{}[]{}[]{} could also do that, but it also uses the same separators for entries and postnotes.  Question 2: How could I change the separator globally for those number-citations from comma to semicolon in biblatex? Gimp: how to remove background noise/artifacts and enhance handwritten text I have a lots of scanned handwritten lecture notes (PDFormat) and I want to improve their readability (I have uploaded an example). In particular I'd like to remove the "squares" on the back. I would appreciate any advice you may have. Thanks in advance. Probability that the sum of 6 dice rolls is even How to remove "phone style" Gnome swipe to unlock I always use align in my documents, and avoid equation. Is there anything wrong with that? My reasoning behind this: align > equation, so why not use it? an "X is an X is an X" what does this mean? I am reading  (I must say, with great delight but that's not relevant) and  Canadian is a Canadian is a Canadian A citizen is a citizen is a citizen.  I do not understand this structure. Why not just &quot;A citizen is a citizen&quot;? What is the meaning of repeating this?