论文标题
在有向图中电阻的度量和超级不平等现象
Metric and ultrametric inequalities for resistances in directed graphs
论文作者
论文摘要
考虑一个电路$ g $每个定向边缘$ e $，其中是一个半导体，具有单个电导函数$ y_e^* = f_e（y_e）= y_e^s /μ_e^r $如果$ y_e \ geq 0 $ y_e \ ge q q qe q qe q qe q e \ geq 0 $ and $ y_e^* = 0 $ y_e \ y_e \ y_e \ leq 0 $。 $ e $在这里是指导边缘，$ y_e $是电位差（电压），$ y_e^*$是$ e $的当前，而$μ_e$是$ e $的电阻；此外，$ r $和$ s $是所有边缘共有的两个严格的正面实际参数。尤其是，$ r = s = 1 $对应于欧姆法律，而$ r = \ frac {1} {2}，s = 1 $可以解释为液压和气体动力学典型的电阻方法律。我们将证明，对于电路的每对有序节点$ a，b $，有效的阻力$μ_{a，b} $定义很好。换句话说，任何带有杆$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a a $ a $ a $ a $ y $μ__{a，b} $的两极网络都可以有效地取代。此外，每三个节点$ a，b，c $ $μ_{a，c}^{s/r} +μ_{ MSC课程：11J83，90C25，94C15,94C99
Consider an electrical circuit $G$ each directed edge $e$ of which is a semiconductor with a monomial conductance function $y_e^* = f_e(y_e) = y_e^s / μ_e^r$ if $y_e \geq 0$ and $y_e^* = 0$ if $y_e \leq 0$. Here $e$ is a directed edge, $y_e$ is the potential difference (voltage), $y_e^*$ is the current in $e$, and $μ_e$ is the resistance of $e$; furthermore, $r$ and $s$ are two strictly positive real parameters common for all edges. In particular, case $r = s = 1$ corresponds to the Ohm law, while $r = \frac{1}{2}, s =1$ may be interpreted as the square law of resistance typical for hydraulics and gas dynamics. We will show that for every ordered pair of nodes $a, b$ of the circuit, the effective resistance $μ_{a,b}$ is well-defined. In other words, any two-pole network with poles $a$ and $b$ can be effectively replaced by two oppositely directed edges, from $a$ to $b$ of resistance $μ_{a,b}$ and from $b$ to $a$ of resistance $μ_{b,a}$. Furthermore, for every three nodes $a, b, c$ the inequality $μ_{a,c}^{s/r} + μ_{c,b}^{s/r} \geq μ_{a,b}^{s/r}$ holds, in which the equality is achieved if and only if every directed path from $a$ to $b$ contains $c$. MSC classes: 11J83, 90C25, 94C15,94C99
