[Search for users] [Overall Top Noters] [List of all Conferences] [Download this site]

Conference rusure::math

Title:	Mathematics at DEC

Moderator:	RUSURE::EDP

Created:	Mon Feb 03 1986
Last Modified:	Fri Jun 06 1997
Last Successful Update:	Fri Jun 06 1997
Number of topics:	2083
Total number of notes:	14613

1943.0. "Math Magazine 1467" by RUSURE::EDP (Always mount a scratch monkey.) Thu Feb 23 1995 12:51

    Proposed by John A. Baker, University of Waterloo, Waterloo, Ontario,
    Canada.
    
    Suppose that v[0], v[1], ..., v[n] are the vertices of a regular
    simplex, S, in R^n centered at the origin.  Let
    
    	v[i] = (v[i,1], v[i,2], ..., v[i,n]) for 0 <= i <= n.
    
    Prove that, for some c > 0,
    
    	sum(v[i,j]^2, i=0..n) = c for all j = 1, 2, ..., n.

T.R Title User Personal
Name Date Lines

1943.1 squares of nth co-ordinates of simplex vertices HERON::BUCHANAN Et tout sera bien et Fri Mar 03 1995 12:27 40

T.R	Title	User	Personal Name	Date	Lines
1943.1	squares of nth co-ordinates of simplex vertices	HERON::BUCHANAN	Et tout sera bien et	`Fri Mar 03 1995 12:27`	40
	> Suppose that v[0], v[1], ..., v[n] are the vertices of a regular > simplex, S, in R^n centered at the origin. Let > > v[i] = (v[i,1], v[i,2], ..., v[i,n]) for 0 <= i <= n. > > Prove that, for some c > 0, > > sum(v[i,j]^2, i=0..n) = c for all j = 1, 2, ..., n. Let the matrix V be defined as: (V[0,1] V[0,2] ... V[0,n]) (V[1,1] V[1,2] ... V[1,n]) ( . . . ) ( . . . ) ( . . . ) (V[n,1] V[n,2] ... V[n,n]) Then we are interested in the value of the diagonal elements of V'.V (where ' denotes transposition.) But what we know is: (V.V')[j1,j2] = K^2 if j1=j2 K^2.d if j1<>j2 where K is the distance of each point from the origin, and d is the cos of the angle the two vertices subtend at the origin. Now, append an extra column to the left of V, so that U = C\|V. U[i,j] = V[i,j] if j > 0 = Ksqrt(-d) if j=0 Then by construction, U.U' = K^2(1-d)I (I is the identity matrix.) So U'.U = U.U'. U'.U[0,0] = -nK^2d => d = -1/(n-1). So c = K^2n/(n-1). Andrew.

>    Suppose that v[0], v[1], ..., v[n] are the vertices of a regular
>    simplex, S, in R^n centered at the origin.  Let
>    
>    	v[i] = (v[i,1], v[i,2], ..., v[i,n]) for 0 <= i <= n.
>    
>    Prove that, for some c > 0,
>    
>    	sum(v[i,j]^2, i=0..n) = c for all j = 1, 2, ..., n.

	Let the matrix V be defined as:

	(V[0,1] V[0,2] ... V[0,n])
	(V[1,1] V[1,2] ... V[1,n])
	(   .	   .	      .  )
	(   .	   .	      .  )
	(   .	   .	      .  )
	(V[n,1] V[n,2] ... V[n,n])

	Then we are interested in the value of the diagonal elements of V'.V
(where ' denotes transposition.) But what we know is:

	(V.V')[j1,j2] = K^2   if j1=j2
		        K^2.d if j1<>j2

	where K is the distance of each point from the origin, and d is the cos
of the angle the two vertices subtend at the origin.

	Now, append an extra column to the left of V, so that U = C|V.
	
	U[i,j] = V[i,j]   if j > 0
	       = K*sqrt(-d) if j=0

	Then by construction, U.U' = K^2*(1-d)*I (I is the identity matrix.)
So U'.U = U.U'.

	U'.U[0,0] = -n*K^2*d => d = -1/(n-1).

	So c = K^2*n/(n-1).

Andrew.