Multiplexed imaging allows multiple cell types to be simultaneously visualised in a single tissue sample, generating unprecedented amounts of spatially-resolved, biological data. In topological data analysis, persistent homology provides multiscale descriptors of "shape" suitable for the analysis of such spatial data. Here we propose a novel visualisation of persistent homology (PH) and fine-tune vectorisations thereof (exploring the effect of different weightings for persistence images, a prominent vectorisation of PH). These approaches offer new biological interpretations and promising avenues for improving the analysis of complex spatial biological data especially in multiple cell type data. To illustrate our methods, we apply them to a lung data set from fatal cases of COVID-19 and a data set from lupus murine spleen.

The end-to-end speech freeze-out fraction formula is derived in a multiple-link network employing a network-wide simultaneous speech interpolation (e.g., burst-switched network). The freeze-out fraction formula for a TASI system derived by Weinstein is shown to be a special case of this formula. A computing algorithm for the formula is developed for a tandem network in which calls can arise between any pair of nodes. To overcome the complexity of the algorithm, which is exponential with respect to the number of originating nodes in the network, an approximate computing algorithm is developed. Accuracy of the approximate algorithm is validated through both theoretical arguments and numerical examples. An extension of the algorithm to other networks such as ring, tree, and mesh is also considered.