Topological invariants for projection method patterns

Topological equivalence to a projection

We present a necessary and sufficient condition for the topological equivalence of a continuous function on a plane to a projection onto one of coordinates.

Cohomology groups for projection point patterns

Aperiodic point sets (or tilings) which can be obtained by the method of cut and projection from higher dimensional periodic sets play an important role for the description of quasicrystals. Their topological invariants can be computed using the higher dimensional periodic structure. We report on the results obtained for the cohomology groups of projection point patterns supplemented by explicit calculations made by F. Gaehler for many well-known icosahedral tilings.

Integral cohomology of rational projection method patterns

We study the cohomology and hence $K$-theory of the aperiodic tilings formed by the so called 'cut and project' method, i.e., patterns in $d$ dimensional Euclidean space which arise as sections of higher dimensional, periodic structures. They form one of the key families of patterns used in quasicrystal physics, where their topological invariants carry quantum mechanical information. Our work develops both a theoretical framework and a practical toolkit for the discussion and calculation of their integral cohomology, and extends previous work that only successfully addressed rational cohomolog…

COVID-19疫情期间死亡率模式的拓扑数据分析

Topological Data Analysis of Mortality Patterns During the COVID-19 Pandemic

本文利用拓扑数据分析中的持续同调和MAPPER方法，分析美国CDC提供的2020年1月至2023年9月疫情期间每周死亡率数据，通过整体及分区域考察数据形状，并结合疫情事件解释观察到的模式。

Deep Invertible Approximation of Topologically Rich Maps between Manifolds

How can we design neural networks that allow for stable universal approximation of maps between topologically interesting manifolds? The answer is with a coordinate projection. Neural networks based on topological data analysis (TDA) use tools such as persistent homology to learn topological signatures of data and stabilize training but may not be universal approximators or have stable inverses. Other architectures universally approximate data distributions on submanifolds but only when the latter are given by a single chart, making them unable to learn maps that change topology. By exploiting…

Robust Detection of Periodic Patterns in Gene Expression Microarray Data using Topological Signal Analysis

In this paper, we present a new approach for analyzing gene expression data that builds on topological characteristics of time series. Our goal is to identify cell cycle regulated genes in micro array dataset. We construct a point cloud out of time series using delay coordinate embeddings. Persistent homology is utilized to analyse the topology of the point cloud for detection of periodicity. This novel technique is accurate and robust to noise, missing data points and varying sampling intervals. Our experiments using Yeast Saccharomyces cerevisiae dataset substantiate the capabilities of the …