AI 中文总结
本文针对超端口网络的缺失全子式情况，确定了生成森林的符号规则，通过完备商关联行列式给出任意响应子式的计算式，简化了雅可比因子，明确了非零条件。
AI 中文摘要
电网络可通过响应矩阵在边界处被概括：规定的边界电压决定边界电流。超端口网络通过将边界端子分组为超端口添加约束，要求每组内总电流为零，且组内电压差为自然坐标。早期工作确定了单个响应项的森林公式以及整个响应矩阵的行列式，缺失的情况是任意子式（即 minors）：不仅需要知道哪些生成森林有贡献，还需要知道每个森林携带的符号。本文提供了该符号规则。在每个超端口中选择一个参考顶点后，响应为 \\(L=(D^{T}K^{-1}D)^{-1}\\)，其中 \\(K\\) 是接地加权拉普拉斯矩阵，\\(D\\) 记录所选电压差。对物理生成森林 \\(F\\) 的分量进行收缩，会得到一个小得多的商端口图 \\(H_F\\)。其简约关联矩阵 \\(B_F=Q_FD\\) 的列仅为 \\(0\\)、\\(\pm e_a\\)、\\(e_a-e_b\\) 形式，因此每个平方关联子式恰好为 \\(0\\) 或 \\(\pm1\\)。对于大小为 \\(k\\) 的响应坐标集 \\(I\\)，将选择器行 \\(E_I^{T}\\) 附加到 \\(B_F\\) 上，定义完备商关联行列式 \\(\widehat{\chi}_F(I)=\det\begin{pmatrix} B_F\\\\ E_I^{T} \end{pmatrix}\\)。对于大小相同的坐标集 \\(I,J\\)，任意响应子式是加权生成森林和，其系数仅为 \\(\widehat{\chi}_F(I)\widehat{\chi}_F(J)\\)，因此推导中使用的雅可比互补子式因子从最终定理中消失。直接块三角化简可得 \\(\widehat{\chi}_F(I)\in\{0,\pm1\}\\)，且非零当且仅当互补商边 \\(N\setminus I\\) 形成 \\(H_F\\) 的生成树。
英文摘要
An electrical network can be summarized at its boundary by a response matrix: prescribed boundary voltages determine boundary currents. A superport network adds a constraint by grouping boundary terminals into superports, requiring the total current in each group to be zero and making voltage differences inside the groups the natural coordinates. Earlier work determined forest formulas for a single response entry and for the determinant of the whole response matrix. The missing case was an arbitrary subdeterminant, or minor: one needs to know not only which spanning forests contribute, but also the sign carried by each forest. This paper supplies that sign rule. After choosing one reference vertex in each superport, the response is \[ L=\left(D^{T}K^{-1}D\right)^{-1}, \] with \(K\) the grounded weighted Laplacian and \(D\) recording the selected voltage differences. Contracting the components of a physical spanning forest \(F\) produces a much smaller quotient port graph \(H_F\). Its reduced incidence matrix \[ B_F=Q_FD \] has columns only of the forms \[ 0,\qquad \pm e_a,\qquad e_a-e_b. \] Hence every square incidence minor is exactly \(0\) or \(\pm1\). For a \(k\)-set of response coordinates \(I\), append to \(B_F\) the selector rows \(E_I^{T}\) and define the completed quotient-incidence determinant \[ \widehatχ_F(I) = \det \begin{pmatrix} B_F\\ E_I^{T} \end{pmatrix}. \] For coordinate sets \(I,J\) of the same size, the arbitrary response minor is a weighted spanning-forest sum whose coefficient is simply \[ \widehatχ_F(I)\widehatχ_F(J). \] Thus the Jacobi complementary-minor factors used in the derivation disappear from the final theorem. Direct block-triangular reduction gives \[ \widehatχ_F(I)\in\{0,\pm1\}, \] with nonvanishing occurring exactly when the complementary quotient edges \(N\setminus I\) form a spanning tree of \(H_F\).
Comments11 pages, 7 figures, 2 tables
