Hypercube Line Picking
المؤلف:
Finch, S. R
المصدر:
"Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press
الجزء والصفحة:
...
20-8-2018
3343
Hypercube Line Picking
Let two points 
and 
be picked randomly from a unit 
-dimensional hypercube. The expected distance between the points 
is then
 |
(1)
|
This multiple integral has been evaluated analytically only for small values of 
. The case 
corresponds to the line line picking between two random points in the interval ![[0,1]](http://mathworld.wolfram.com/images/equations/HypercubeLinePicking/Inline7.gif)
.
The first few values for 
are given in the following table.
 |
Sloane |
 |
| 1 |
-- |
0.3333333333... |
| 2 |
A091505 |
0.5214054331... |
| 3 |
A073012 |
0.6617071822... |
| 4 |
A103983 |
0.7776656535... |
| 5 |
A103984 |
0.8785309152... |
| 6 |
A103985 |
0.9689420830... |
| 7 |
A103986 |
1.0515838734... |
| 8 |
A103987 |
1.1281653402... |
The function 
satisfies
![1/3n^(1/2)<=Delta(n)<=(1/6n)^(1/2)sqrt(1/3[1+2(1-3/(5n))^(1/2)])](http://mathworld.wolfram.com/images/equations/HypercubeLinePicking/NumberedEquation2.gif) |
(2)
|
(Anderssen et al. 1976), plotted above together with the actual values.
M. Trott (pers. comm., Feb. 23, 2005) has devised an ingenious algorithm for reducing the 
-dimensional integral to an integral over a 1-dimensional integrand 
such that
 |
(3)
|
The first few values are
In the limit as 
, these have values for 
, 2, ... given by 
times 2/3, 6/5, 50/21, 38/9, 74/11, ... (OEIS A103990 and A103991).
This is equivalent to computing the box integral
![Delta_n(s)=s/(Gamma(1-1/2s))int_0^infty(1-[d(u)]^n)/(u^(s+1))du](http://mathworld.wolfram.com/images/equations/HypercubeLinePicking/NumberedEquation4.gif) |
(8)
|
where
(Bailey et al. 2006).
These give closed-form results for 
, 2, 3, and 4:
where 
is a Clausen function, 
is Catalan's constant, and
 |
(16)
|
The 
case above seems to be published here for the first time; the simplified form given above is due to Bailey et al. (2006). Attempting to reduce 
to quadratures gives closed-form pieces with the exception of the single piece
which appears to be difficult to integrate in closed form (Bailey et al. 2007, p. 272).
The value 
obtained for cube line picking is sometimes known as the Robbins constant.
REFERENCES:
Anderssen, R. S.; Brent, R. P.; Daley, D. J.; and Moran, A. P. "Concerning 
and a Taylor Series Method." SIAM J. Appl. Math. 30, 22-30, 1976.
Bailey, D. H.; Borwein, J. M.; and Crandall, R. E. "Box Integrals." Preprint. Apr. 3, 2006.
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action.Wellesley, MA: A K Peters, p. 272, 2007.
Finch, S. R. "Geometric Probability Constants." §8.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 479-484, 2003.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, p. 30, 1983.
Robbins, D. "Average Distance between Two Points in a Box." Amer. Math. Monthly 85, 278, 1978.
Sloane, N. J. A. Sequences A073012, A091505, A103983, A103984, A103985, A103986, A103987, A103988, A103989,A103990, and A103991 in "The On-Line Encyclopedia of Integer Sequences."
Trott, M. "The Area of a Random Triangle." Mathematica J. 7, 189-198, 1998.
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