The statement, "Suppose that R is fixed. Then the volume of an n-ball of
radius R approaches zero as n tends to infinity." is in the link that Raul
mentioned in his second post in this thread together with a couple of
proofs.
A related statement (assuming I am not mistaken) is: Given a fixed R
(re
On Wed, Aug 16, 2017 at 10:12 AM, Murray Eisenberg
wrote:
...
> Of course, an exact formula for the n-dimensional measure of the unit n-ball
> is known:
>
> Pi^(n/2)
> V(n) = --
> Gamma(1+n/2)
Which brings us back to the first line of
P.S. If with Mathematica you do want to see decimal approximations, after the
input/output shown, do it with:
N[%] (* % is preceding output *)
{2., 3.14159, 4.18879, 4.9348, 5.26379, 5.16771, 4.72477, 4.05871, 3.29851,
2.55016}
> On 16 Aug2017, at 10:12 AM, Murray Eisenberg wrote:
>
On Tuesday, August 15, 2017 4:27:20 PM EDT Raul Miller wrote:
> It loses accuracy somewhere between n=150 and n=200.
>
> Keep in mind though that the dimensions of these "volumes" are not
> comparable.
That's a good point. Observing alone that the volumes of the spheres increase
and then decrease
bill lam wrote:
> Has the n-sphere become concave in higher dimension?
>
Think of it this way: you're nesting a sphere inside a box, so the volume
"wasted" is simply the corners of the box. But every time you increase the
number of dimensions, you vastly increase the number of corners the box ha
Actually _ stands for positive infinity
1%0
_
_1%0
__
when Roger Hui and Ken Iverson designed J, their mathematical perspective
built upon previous APL experiences. For example, 0%0 became 0 in J whereas
it is 1 in APL. E. E. McDonnell wrote a thorough article documenting this
choice:
http
Why get decimal approximations when you can get the exact values?
With Mathematica, for example, one finds:
Table[RegionMeasure[Ball[n]], {n, 1, 10}]
{2, Pi, (4 Pi)/3, Pi^2/2, (8 Pi^2)/15, Pi^3/6, (16 Pi^3)/105, Pi^4/24, (32
Pi^4)/945, Pi^5/120}
(Actually, the output from Mathematica gives
Thanks Jimmy. When I translated you answer into J, I got the same result. But,
if the csc of a number isundefined, should the signum be 1?
load 'plot'
do=: 13 : '({.y) +(i.{:y)*(--/ 2{.y)%<:{:y'
A=:do 0 2p1 17
]csc=:*1%1 o. A
1 1 1 1 1 1 1 1 1 _1 _1 _1 _1 _1 _1 _1 _1
]sec=:*1%2
Hi Bill, I have just been playing with a circle and and a hypocycoid, and
imploding a circle:
load '~addons/graphics/fvj4/dwin.ijs'
do=: 13 :'({.y) +(i.{:y)*(--/ 2{.y)%<:{:y'
circle=:|: 2 1 o. /t=:do 0 2p1 81
_1 _1 1 1 dwin 'dwin polar circle'
200 0 200 dpoly circle
hypo=:|:(2 1 o./t=:do 0 2p1
Ok, so.. thinking this through.
The offsets you are interested in at any dimension are positive
integer multiples }.0 1 _1. (The strangeness with the zero should make
sense in a moment.)
For higher dimension n, you are interested the coordinate offsets you
are interested in are positive integer m
As Jimmy Gauvin has already pointed out; no, you should not expect
zeros from secant nor cosecant. If you graph them, you should see why.
(The minimum absolute value of either of those will be 1.)
Thanks,
--
Raul
On Tue, Aug 15, 2017 at 8:13 PM, Linda Alvord wrote:
> Thanks Raul. It shows th
Yes.
(That said, I've not tried coding up that simulation. If we simulate
points in a unit n-cube, discarding those outside the unit sphere,
this quickly becomes inefficient for higher dimensions. If we use some
other technique, though, we run into the problem of showing that the
distribution is v
Is the distance from the origin
%:+/*: y
can we use discrete points simulation to verify
the number of points satisfying the inequality
R>:%:+/*: y
is actually diminishing for large n?
Вт, 15 авг 2017, Jimmy Gauvin написал(а):
> The construction of the sphere implies it cannot be convex but you
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