duvin103time
New member
- Joined
- Jun 14, 2006
- Messages
- 9
Let Vr = ( exp(j*kr'*P1) exp(j*kr'*P2) ... exp(j*kr'*Pn) )'
where ' denotes Hermitian, kr (r=1,2,...,m) and Pi (i=1,2,...,n) are 3X1 real vectors, j is imaginary unit, exp is exponent.
Let G(V1,V2,...,Vm) denotes the Gram determinant of the vetors V1, V2,..., Vm, i.e.
G(V1,V2,...,Vm) = det[<Vr,Vs>] (1<=r,s<=m)
where <Vr,Vs>=Vr'*Vs.
We define
Dm = G(V1,V2,...,Vm+1)/G(V1,V2,...,Vm)
and
Int(Dm,dK) = int(int( ... int(int(Dm,dk1),dk2), ... dkm),dkm+1)
where int denotes integral on the unit sphere.
Consider m=1,
D1 = [n^2-|<V1,V2>|^2]/n
Int(D1,dK) = [n^2-(n+(sinA/A)^2)]/n <= n-1
I wonder if
Int(Dm,dK) <= n-m when m>=2
The conjecture indicates the averaged distance between the nX1 vector Vm+1 and the linear space spanned by V1,V2,...,Vm is less than n-m.
Numberical results shows my conjecture must be true. First, generate a group of p1, p2,...,pn randomly. Second, generate many groups of k1,k2,...,km,km+1 randomly located on the unit sphere. Third, compute Dm respectively. Finally, the mean value of Dm is found for p1,p2,...pn. Repeat the steps mentioned above, we will get many mean values of Dm for groups of p1,p2,...,pn. All of them are less than n-m. But I cannot prove the conjecture.
Many thanks!
where ' denotes Hermitian, kr (r=1,2,...,m) and Pi (i=1,2,...,n) are 3X1 real vectors, j is imaginary unit, exp is exponent.
Let G(V1,V2,...,Vm) denotes the Gram determinant of the vetors V1, V2,..., Vm, i.e.
G(V1,V2,...,Vm) = det[<Vr,Vs>] (1<=r,s<=m)
where <Vr,Vs>=Vr'*Vs.
We define
Dm = G(V1,V2,...,Vm+1)/G(V1,V2,...,Vm)
and
Int(Dm,dK) = int(int( ... int(int(Dm,dk1),dk2), ... dkm),dkm+1)
where int denotes integral on the unit sphere.
Consider m=1,
D1 = [n^2-|<V1,V2>|^2]/n
Int(D1,dK) = [n^2-(n+(sinA/A)^2)]/n <= n-1
I wonder if
Int(Dm,dK) <= n-m when m>=2
The conjecture indicates the averaged distance between the nX1 vector Vm+1 and the linear space spanned by V1,V2,...,Vm is less than n-m.
Numberical results shows my conjecture must be true. First, generate a group of p1, p2,...,pn randomly. Second, generate many groups of k1,k2,...,km,km+1 randomly located on the unit sphere. Third, compute Dm respectively. Finally, the mean value of Dm is found for p1,p2,...pn. Repeat the steps mentioned above, we will get many mean values of Dm for groups of p1,p2,...,pn. All of them are less than n-m. But I cannot prove the conjecture.
Many thanks!