sax 
PONTIFICIAE ACADEMIAE SCIENTIARVM SCRIPTA VARIA - 28 
AL 
(1=1, ..., n). 
That is, the characteristic numbers of the whole system appear 
in pairs with their reciprocals. 
We now make the following assumption which is the joker 
allowing us to avoid cyclic exceptions to the Turnpike 
Theorem; that is, all characteristic numbers X; other than 
\i=2X,.1 have absolute value different from unity. We also 
assume, for simplicity, that 1 is the sole characteristic number 
which is multiple. It can be shown that the solutions to (22) 
and (29) may be written in the following forms: 
b, = Ty + XA + +a, A 
(32) 
z,=[I+D+ … + D] «+o, + vee +a, kA, + 
+BEnn1 + BaEn rad + ee FB.E A, 
where 
D=(1+7) [1 - (A0 + B°+C°)]-"(1 - d) (BOY, 
and ©; ({=1,...,n) and §, (¢=1, .., 2n) are n-dimensional 
column vectors; in particular, €, ; is the eigen-vector of D asso- 
ciated with X,,;  Scalars, &;, > Xp» By» ---> By Can be de- 
termined bv 2# equations: 
zy =k + ob + + ok + BE + Bb t oo +86. 
wrt 
ZT > 
1+D+...+DT Ja,ËE, + ab AT +. +E ATH 
+BE,1 + BaEnnad 142 + FBLA 
9] Morishima - pag. 16