Microstructure evolution of Solid Oxide Fuel Cell anodes characterized by persistent homology

Uncovering microstructure evolution mechanisms that accompany the long-term operation of solid oxide fuel cells is a fundamental challenge in designing a more durable energy system for the future.To date,the study of fuel cell stack degradation has focused mainly on electrochemical performance and,more rarely,on averaged microstructural parameters.Here we show an alternative approach in which an evolution of three-dimensional microstructural features is studied using electron tomography coupled with topological data analysis.The latter produces persistent diagrams of microstructure before and after long-term operation of electrodes.Those diagrams unveil a new insight into the degradation process of three involved phases:Nickel,pores,and yttrium-stabilized zirconium.

Computational Tools in Weighted Persistent Homology

In this paper, the authors study further properties and applications of weighted homology and persistent homology. The Mayer-Vietoris sequence and generalized Bockstein spectral sequence for weighted homology are introduced. For applications, the authors show an algorithm to construct a filtration of weighted simplicial complexes from a weighted network. They also prove a theorem to calculate the mod p^(2) weighted persistent homology provided with some information on the mod p weighted persistent homology.

Persistent Homology for the Quantitative Analysis of the Structure and Stability of Carboranes

Persistent homology is a powerful and novel tool for quantifying the inherent topological features of structure. In this work, we used the persistent homology for the first time to study the closo-carboranes C2Bn-2Hn(n = 5~20) and their parent structures closo-boranes dianions BnHn2-(n = 5~20), where multiple elements are present. All these structures are first investigated with the standard Vitoris-Rips complex. We interpret all barcodes representation and associate them with structural details. By means of average bar length, a linear regression model was established to construct the relationship between persistent homology features and molecular stability, which was expressed by the relative energies. For closo-boranes dianions, we only use B atom set since B and H atoms are in pairs. The average lengths of β0, β1 and β2 bars are used as the features for linear regression, and excellent correlation coefficient(0.977) between the values predicted by persistent homology and those by quantum calculations was achieved. For closo-carboranes, C–B atom set(ignore the differences in the atoms), B atom set and C atom set were considered to get the persistent homology features(since there were only two C atoms in C2Bn-2Hn, only β0 bars were considered), and seven average bar lengths were calculated, respectively. Pearson coefficient of 0.937 was obtained. We found that the stability of carboranes showed a high linear correlation with the characteristics generated from topological bars in H0, H1 and H2. The results show that the topological information generated by persistent homology can be extended and applied to multi-element systems.

Analyzing architectural diversity in maize plants using the skeletonimage-based method

Shoot architecture in maize is critical since it determines resource use,impacts wind and rain damage tolerance,and affects yield stability.Quantifying the diversity among inbred lines in heterosis breeding is essential,especially when describing germplasm resources.However,traditional geometric description methods oversimplify shoot architecture and ignore the plant’s overall architecture,making it difficult to reflect and illustrate diversity.This study presents a new method to describe maize shoot architecture and quantifies its diversity by combining computer vision algorithms and persistent homology.Our results reveal that persistent homology can capture key characteristics of shoot architecture in maize and other details often overlooked by traditional geometric analysis.Based on this method,the morphological diversity of shoot architecture can be mined(quantified),and the main shoot architecture types can be obtained.Consequently,this method can easily describe the diversity of shoot architecture in many maize materials.

Multi-scale visual analysis of cycle characteristics in spatially-embedded graphs

We present a visual analysis environment based on a multi-scale partitioning of a 2d domain intoregions bounded by cycles in weighted planar embedded graphs.The work has been inspired by anapplication in granular materials research,where the question of scale plays a fundamental role inthe analysis of material properties.We propose an efficient algorithm to extract the hierarchical cyclestructure using persistent homology.The core of the algorithm is a filtration on a dual graph exploitingAlexander’s duality.The resulting partitioning is the basis for the derivation of statistical properties thatcan be explored in a visual environment.We demonstrate the proposed pipeline on a few syntheticand one real-world dataset.

Biomolecular Topology:Modelling and Analysis

With the great advancement of experimental tools,a tremendous amount of biomolecular data has been generated and accumulated in various databases.The high dimensionality,structural complexity,the nonlinearity,and entanglements of biomolecular data,ranging from DNA knots,RNA secondary structures,protein folding configurations,chromosomes,DNA origami,molecular assembly,to others at the macromolecular level,pose a severe challenge in their analysis and characterization.In the past few decades,mathematical concepts,models,algorithms,and tools from algebraic topology,combinatorial topology,computational topology,and topological data analysis,have demonstrated great power and begun to play an essential role in tackling the biomolecular data challenge.In this work,we introduce biomolecular topology,which concerns the topological problems and models originated from the biomolecular systems.More specifically,the biomolecular topology encompasses topological structures,properties and relations that are emerged from biomolecular structures,dynamics,interactions,and functions.We discuss the various types of biomolecular topology from structures(of proteins,DNAs,and RNAs),protein folding,and protein assembly.A brief discussion of databanks(and databases),theoretical models,and computational algorithms,is presented.Further,we systematically review related topological models,including graphs,simplicial complexes,persistent homology,persistent Laplacians,de Rham-Hodge theory,Yau-Hausdorff distance,and the topology-based machine learning models.

The Realization Problem for Discrete Morse Functions on Trees

We introduce a new notion of equivalence of discrete Morse functions on graphs called persistence equivalence.Two functions are considered persistence equivalent if and only if they induce the same persistence diagram.We compare this notion of equivalence to other notions of equivalent discrete Morse functions.Then we compute an upper bound for the number of persistence equivalent discrete Morse functions on a fixed graph and show that this upper bound is sharp in the case where our graph is a tree.This is a version of the"realization problem"of the persistence map.We conclude with an example illustrating our construction.