Katsura

The system of equations is of the form:
ì
ï
ï
ï
ï
í
ï
ï
ï
ï
î
um-
N
S
i=-N
uium-i ; m=-N+1..N+1
1-
N
S
i=-N
ui
u-m=um ; m=1..2N-1
um=0 ; m=N+1..2N-1

These equations appeared in a problem of magnetism in physics, which our group studied several years ago. We consider the random Ising model (mixture of the ferro- and the antiferro- magnetic bonds) on the infinite Cayley tree of the coordination number z. The distribution function g(x) (-1£ x£ 1) of the effective field x is a function of the temperature, T, the magnetic field, H, and the concentration of the ferromagnetic bond, p. This distribution function obeys the integral equation (1):
g(x)= ó
õ
 


 
[d(x-T arcth(th(
J
T
)th(
H'
T
)) d(H'-h'-h")
g(h')g(h")(p d(J-1)+(1-p)d(J+1))dH'dh'dh"dJ
When z=3, H=0, p=1/2, and T=0, u(l)=g(Nx,x=l/n) is shown to be the solution of a well precised system of algebraic equations. The general form of the equations1 is the following one (2):
N
S
I=-N
u(l)u(m-l)=u(m)     
N
S
I=-N
u(l)=1
with mÎ{-N+1,...,N-1}, u(l)=u(-l), and u(l)=0 for l>N. The derivation of these equations and the properties of the solutions are given also in the previously quoted paper. The number of solutions of for a given N is 2N. Among them, physically meaningful solutions are restricted to those for which u(l) is real and 0£ u(l)£ 1 (since u(l) is a probability).

The physically acceptable solutions include:
  1. u(0)=1 and all other u(l)=0.
  2. u(-N)=u(0)=u(N)=1/3 and all other u(l)=0.
  3. if N is not a prime and M is a divisor of N then the u(l)'s for such N include the u(l)'s for M. For example, when N is even,
    u(± N)=
    22 +1
    14
    		,
    u(± N/2)=
    3-2
    14
    		, u(0)=
    3-2
    7

    and all other u(l)=0.
That is, the number of the physically acceptable solutions is w(N)+1, where w(N) is the number of divisors of N. These equations have been known perhaps through the paper by Boege et al2 before we published our papers. These equations were used in different sources as tests of Gröbner Basis algorithms and also as tests of the speed of several machines: mainframes, work-stations and personal computers. We may list: [7] T. Sasaki and T. Takeshima. J. Info. Process. 12, 371-379, 1989.

[8] H. Kobayashi, S. Moritsugu and P. W. Hogan. ISSAC-88, K1-K5.

[9] S. Moritsugu. Doctoral Thesis at the University of Tokyo, 1989, K2-K5.

[10] A. Giovini, T. Mora, G. Niesi, L. Robbiano and C. Traverso. ISSAC-91, K4.

[11] H. M. Moeller, T. Mora and C. Traverso. ISSAC-92, K3=K4. [12] M. Noro. 1993, K6.

[13] T. Shimoyamaand T. Takeshima. 1993, K3-K8.

[14] H. Murao, H. Kobayashi and T. Fujise. J. Symbolic Comp. 15, 123, K6.

[15] C. Traverso, L. Donati, (1989) ISSAC-89.

[16] J. C. Fauguere, P. Gianni, D. Lazard and T. Mora. J. Symbolic Comp. (1994).

[17] J. C. Fauguere. Doctoral Thesis, Paris 1994.

Many documents and preprints in Japanese in the period 1985--1988 are omitted in this list.

The Gröbner basis of (2) leads
F(u(0))=0,   u(l)=f(u(0))     (3)
where F and f are equations of degrees 2N and 2N-1 in u(0) respectively. From the numerical solution of (2) and (3) we found that u(1), u(2),...,u(N-1) are roughly equal(except u(0) and u(N)) motivating this fact the study on the behaviour of the solutions when N tends to infinity13 14

The solution is given by the function:
g(x)=ad(x) +(b/2)[d(x-1)+d(x+1)]-c(x2-1)
where a, b and c are constants. Several related problems, extensions to the cases p¹ 1/2, T¹ 0 have been also considered15 16 17 . The reviews are given in 13 and 17.
Characteristics:




Example 1:
kat2:=proc(N,m)
 local l,s:s:=-u(m,N):for l from -N to N do s:=s+u(l,N)*u(m-l,N): od:s:
end:

u:=proc(l,N)
if (l<0) then RETURN(u(-l,N)): else if (l>N) then RETURN(0):else
 RETURN(u[l]):  fi: fi:
end:

kat3:=proc(N)
 local l,s: s:=-1: for l from -N to N do s:=s+u(l,N): od: s:
end:

kat :=proc(N)
 local l,m:
 l:=seq(kat2(N,m),m=(-N+1)..(N-1)),kat3(N):
 [op(l)]:
end:

References

1
S. Katsura, W. Fukuda, S. Inawashiro, N. M. Fujiki and R. Gebauer: Cell Biophysics Vol.11 (1987) p.309-319. See Eq.[3.5] and [3.8].
2
W. Boege, R. Gebauer, H. Kredel: J. Symbol. Comp. 1 (1986) 83.
13
S. Katsura, Prog. Theor. Phys. Suppl. 87 (1986) 139.
14
S. Katsura, Physica 141A (1987) 556; 149A (1988) 371.
15
M. Sasaki (Seino) and S. Katsura, Physica A 157 (1990) 1195.
16
M. Seino and S. Katsura, Prog. Theor. Phys. Suppl. 115 (1994) 237.
17
S. Katsura, in "New Trends in Magnetism", ed by M. D. Coutinho-Filho and S. M. Resende (World Scientific, 1990).