Refinement of Interval Approximations for Fully Commutative Quivers

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals. Additionally, it provides a mechanism to tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. For the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. For the latter, we introduce a new invariant that reveals persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We subsequently present several models for constructing commutative ladder filtrations, offering fresh insights into random filtrations and demonstrating our toolkit's effectiveness in analyzing the topology of materials

Fractal mechanism of basin of attraction in passive dynamic walking

Passive dynamic walking is a model that walks down a shallow slope without any control or input. This model has been widely used to investigate how humans walk with low energy consumption and provides design principles for energy-efficient biped robots. However, the basin of attraction is very small and thin and has a fractal-like complicated shape, which makes producing stable walking difficult. In our previous study, we used the simplest walking model and investigated the fractal-like basin of attraction based on dynamical systems theory by focusing on the hybrid dynamics of the model composed of the continuous dynamics with saddle hyperbolicity and the discontinuous dynamics caused by the impact upon foot contact. We clarified that the fractal-like basin of attraction is generated through iterative stretching and bending deformations of the domain of the Poincaré map by sequential inverse images. However, whether the fractal-like basin of attraction is actually fractal, i.e., whether infinitely many self-similar patterns are embedded in the basin of attraction, is dependent on the slope angle, and the mechanism remains unclear. In the present study, we improved our previous analysis in order to clarify this mechanism. In particular, we newly focused on the range of the Poincaré map and specified the regions that are stretched and bent by the sequential inverse images of the Poincaré map. Through the analysis of the specified regions, we clarified the conditions and mechanism required for the basin of attraction to be fractal

Sudden change in fractality of basin boundary in passive dynamic walking

The 11th International Symposium on Adaptive Motion of Animals and Machines. Kobe University, Japan. 2023-06-06/09. Adaptive Motion of Animals and Machines Organizing Committee.Poster Session P7

Flow estimation solely from image data through persistent homology analysis

Abstract: Topological data analysis is an emerging concept of data analysis for characterizing shapes. A state-of-the-art tool in topological data analysis is persistent homology, which is expected to summarize quantified topological and geometric features. Although persistent homology is useful for revealing the topological and geometric information, it is difficult to interpret the parameters of persistent homology themselves and difficult to directly relate the parameters to physical properties. In this study, we focus on connectivity and apertures of flow channels detected from persistent homology analysis. We propose a method to estimate permeability in fracture networks from parameters of persistent homology. Synthetic 3D fracture network patterns and their direct flow simulations are used for the validation. The results suggest that the persistent homology can estimate fluid flow in fracture network based on the image data. This method can easily derive the flow phenomena based on the information of the structure

Structure and properties of densified silica glass: characterizing the order within disorder

世界一構造秩序のあるガラスの合成と構造解析に成功 --ガラスの一見無秩序な構造の中に潜む秩序を抽出--. 京都大学プレスリリース. 2021-12-25.The broken symmetry in the atomic-scale ordering of glassy versus crystalline solids leads to a daunting challenge to provide suitable metrics for describing the order within disorder, especially on length scales beyond the nearest neighbor that are characterized by rich structural complexity. Here, we address this challenge for silica, a canonical network-forming glass, by using hot versus cold compression to (i) systematically increase the structural ordering after densification and (ii) prepare two glasses with the same high-density but contrasting structures. The structure was measured by high-energy X-ray and neutron diffraction, and atomistic models were generated that reproduce the experimental results. The vibrational and thermodynamic properties of the glasses were probed by using inelastic neutron scattering and calorimetry, respectively. Traditional measures of amorphous structures show relatively subtle changes upon compacting the glass. The method of persistent homology identifies, however, distinct features in the network topology that change as the initially open structure of the glass is collapsed. The results for the same high-density glasses show that the nature of structural disorder does impact the heat capacity and boson peak in the low-frequency dynamical spectra. Densification is discussed in terms of the loss of locally favored tetrahedral structures comprising oxygen-decorated SiSi4 tetrahedra