Pattern detection in network models provides insights to both global structure and local node interactions. In particular, studying patterns embedded within remittance and migration flow networks can be useful in understanding economic and sociologic trends and phenomena and their implications both in regional and global settings. We illustrate how topo-algebraic methods can be used to detect both local and global patterns that highlight simultaneous interactions among multiple nodes, giving a more holistic perspective on the network fabric and a higher order description of the overall flow structure of directed networks. Using the 2015 Asian net migration and remittance networks, we build and study the associated directed clique complexes whose topological features correspond to specific flow patterns in the networks. We generate diagrams recording the presence, persistence, and perpetuity of patterns and show how these diagrams can be used to make inferences about the characteristics of migrant movement patterns and remittance flows.

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The problem of finding square roots of p-adic integers in Z p , p 2, has been a classic application of Hensel's lemma. A recent development on this problem is the application and analysis of convergence of numerical methods in approximating p-adic numbers. For a p-adic number a, Zerzaihi, Kecies, and Knapp (2010) introduced a fixedpoint method to find the square root of a in Q p . Zerzaihi and Kecies (2011) later extended this problem to finding the cube root of a using the secant method. In this paper, we compute for the square roots and cube roots of p-adic numbers in Q p , using the Newton-Raphson method. We present findings that confirm recent results on the square roots of p-adic numbers, and highlight the advantages of this method over the fixed point and secant methods. We also establish sufficient conditions for the convergence of this method, and determine the speed of its convergence. Finally, we detemine how many iterations are needed to obtain a specified number of correct digits in the approximate.

This paper accompanies Team Cordi-Ak's entry to the classification of 12-lead ECGs for the Phys-ioNet/Computing in Cardiology Challenge 2020. Our approach leverages mathematically computable topological signatures of 12-lead ECGs as proxy for features informed by medical expertise to train a two-level random forest model in a multi-class classification task. We view ECGs as multivariate time series data and convert different segments and groupings of leads to point cloud embeddings. This stores both local and global structures of ECGs, and encodes periodic information as attractor cycles in high-dimensional space. We then employ topological data analysis on these embeddings to extract topological features based on different summaries available in the literature. We supplement these features with demographic data and statistical moments of RR intervals based on the Pan-Tompkins algorithm for each lead to train the classifier. Our classifier achieved a challenge validation score of 0.304, and a final score of -0.113 on the full hidden test data, placing us 37th out of 41 officially ranked teams that participated in this year's Challenge.