AI 中文总结
本文首次计算了以 SmallGroup(27,3) 为根的后代树中终顶点的实概率分布熵，并比较了 SmallGroup(729,45) 后代树中实与虚测度的熵。
AI 中文摘要
研究以 Delta=SmallGroup(27,3) 为根、具有极大幂零类的 3-群 G 的后代树 T(Delta)，我们首次证明，对于该无限有向图上主线之外的所有终顶点，其实概率分布 P_r(G) 的熵值 H(P_r)=sum P_r(G)|log(P_r(G))| 也可确定。这些顶点作为实二次域 K=Q(d^1/2), d>0 的最大非分歧 pro-3-扩张 F(3,infinity,K) 的自同构群 Gal(F(3,infinity,K)/K) 出现。对于另一棵以 N=SmallGroup(729,45) 为根、coclass 无界的后代树 T(N) 中的群 G（该树已由 L. Bartholdi 和 M. R. Bush 研究），我们将实测度 P_r(G) 的熵 H(P_r) 与虚测度 P_i(G) 的熵 H(P_i) 进行比较。
英文摘要
Investigating the descendant tree T(Delta) of 3-groups G of maximal nilpotency-class, with root Delta=SmallGroup(27,3), we show for the first time that an entropy-value H(P_r)=sum P_r(G)|log(P_r(G))| can also be determined for the real probability-distribution P_r(G) of all terminal vertices outside of the mainline on this infinite digraph. These vertices arise as automorphism groups Gal(F(3,infinity,K)/K) of maximal unramified pro-3-extensions F(3,infinity,K) of real-quadratic number fields K=Q(d^1/2), d>0. For the groups G of another descendant tree T(N), with root N=SmallGroup(729,45) and unbounded coclass, which has been studied by L. Bartholdi and M. R. Bush, we compare the entropy H(P_r) of the real measure P_r(G) with the entropy H(P_i) of the imaginary measure P_i(G).
Comments7 pages, 1 figure, 4 tables
