标题： 使用超度量近似有限多拓扑系统上的扩散 显示英文标题

标题： Approximating Diffusion on Finite Multi-Topology Systems Using Ultrametrics

摘要： 受多拓扑建筑和城市模型数据的启发，首先给出了在给定有限集上通过顶点-边-权重图对多个$T_0$-拓扑的无损表示，并提出了相关加权图距离矩阵的次优超度量作为这些数据的索引结构。 这被应用于一种启发式并行拓扑排序算法，用于边权重有向无环图。 这种结构化数据在分布式处理器上的建筑或城市模型上的热流等过程模拟中具有研究价值。 鉴于此，本文大部分内容计算了与有限图$G$相关的紧开子域的$p$-adic 数域上的某些无界自伴$p$-adic 拉普拉斯算子在$L^2$-空间上的谱，相对于限制的哈尔测度。 以及借助$p$-adic 多项式插值从$G$顶点上的超度量得到的雷登测度。 最后，通过这些算子的有限逼近，给出了相应热方程解的误差界限。 显示英文摘要

摘要： Motivated by multi-topology building and city model data, first a lossless representation of multiple $T_0$-topologies on a given finite set by a vertex-edge-weighted graph is given, and the subdominant ultrametric of the associated weighted graph distance matrix is proposed as an index structure for these data. This is applied in a heuristic parallel topological sort algorithm for edge-weighted directed acyclic graphs. Such structured data are of interest in simulation of processes like heat flows on building or city models on distributed processors. With this in view, the bulk of this article calculates the spectra of certain unbounded self-adjoint $p$-adic Laplacian operators on the $L^2$-spaces of a compact open subdomain of the $p$-adic number field associated with a finite graph $G$ with respect to the restricted Haar measure. as well as to a Radon measure coming from an ultrametric on the vertices of $G$ with the help of $p$-adic polynomial interpolation. In the end, error bounds are given for the solutions of the corresponding heat equations by finite approximations of such operators.